SlideShare una empresa de Scribd logo
1 de 10
Descargar para leer sin conexión
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
165
NEW CHARACTERIZATION OF KERNEL SET IN FUZZY TOPOLOGICAL
SPACES
Qays Hatem Imran
Al-Muthanna University, College of Education, Department of Mathematics, Al-Muthanna, Iraq
Basim Mohammed Melgat
Foundation of Technical Education, Technical Institute of Al-Mussaib, Babylon, Iraq
ABSTRACT
In this paper we introduce a kernelled fuzzy point, boundary kernelled fuzzy point and
derived kernelled fuzzy point of a subset A of X, and using these notions to define kernel set of fuzzy
topological spaces. Also we introduce fuzzy topological ݇‫-ݎ‬ space. The investigation enables us to
present some new fuzzy separation axioms between ‫ܶܨ‬଴ and ‫ܶܨ‬ଵ- spaces.
Keywords: Fuzzy Topological Space, Kernelled Fuzzy Point, Boundary Kernelled Fuzzy Point,
Derived Kernelled Fuzzy Point, Kernel Set, Weak Fuzzy Separation Axioms, ‫ܴܨ‬௜- space, ݅ ൌ 0,1.
1. INTRODUCTION
The concept of fuzzy set and fuzzy set operations were first introduced by L. A. Zadeh in
1965 [8]. After Zadeh 's introduction of fuzzy sets, Chang [3] defined and studied the notion of fuzzy
topological space in 1968. In 1997, fuzzy generalized closed set (Fg - closed set) was introduced by
G. Balasubramania and P. Sundaram [6]. In 1998, the notion of Fgs - closed set was defined and
investigated by H. Maki el al [7]. In 2002, O. Bedre Ozbakir [11], defined the concept of fuzzy
generalized strongly closed set. In 1984, fuzzy separation axioms have been introduced and
investigated by A. S. Mashour and others [1].
In this paper we introduce a new characterization of kernel set through our definition
kernelled fuzzy point, boundary kernelled fuzzy point and derived kernelled fuzzy point. By these
notions, we obtain that the kernel of a set in fuzzy topological space ሺܺ, ܶሻ is a union of the set itself
with the set of all boundary kernelled fuzzy points. In addition, it is a union of the set itself with the
set of all derived kernelled fuzzy points and we give some result of ‫ܴܨ‬଴- space by using these
notions. Also in this paper we introduce fuzzy topological ݇‫-ݎ‬ space iff kernel of a subset A of X is
INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
TECHNOLOGY (IJCET)
ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 5, Issue 4, April (2014), pp. 165-174
© IAEME: www.iaeme.com/ijcet.asp
Journal Impact Factor (2014): 8.5328 (Calculated by GISI)
www.jifactor.com
IJCET
© I A E M E
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
166
an fuzzy open set. Via this kind of fuzzy topological space, we give a new characterization of fuzzy
separation axioms lying between ‫ܶܨ‬଴ and ‫ܶܨ‬ଵ- spaces.
2. PRELIMINARIES
Fuzzy sets theory, introduced by Lotfi. A. Zada in 1965 [8], is the extension of classical set
theory by allowing the membership of elements to range from 0 to 1. Let X be the universe of a
classical set of objects. Membership in a classical subset A of X is often viewed as a characteristic
function µA from X into {0, 1}, where
µA(‫)ݔ‬ =
for any ‫∈ݔ‬ X.
{0,1} is called a valuation set (see [13]). If the valuation set is allowed to be the real interval [0,1], A
is called a fuzzy set in X. µA(‫)ݔ‬ (or simply A(‫))ݔ‬ is the membership value (or degree of membership)
of ‫ݔ‬ in A. Clearly, A is a subset of X that has
no sharp boundary. A fuzzy set A in X can be represented by the set of pairs: A ={( ‫ݔ‬ , A(‫,))ݔ‬ ‫∈ݔ‬ X}.
Let A : X → [0,1] be a fuzzy set. If A(‫)ݔ‬ = 1, for each ‫∈ݔ‬ X, we denote it by 1X and if A(‫)ݔ‬ = 0, for
each ‫∈ݔ‬ X, we denote it by 0X . That is, by 0X and 1X , we mean the constant fuzzy sets taking the
values 0 and 1 on X, respectively [2]. Let Ι =[0,1]. The set of all fuzzy sets in X, denoted by ΙX
[10].
Definition 2.1:[4] Let A be a fuzzy set of a set X. The support of A is the elements ‫ݔ‬ whose
membership value is greater than 0, i.e., supp(A) = { ‫∈ݔ‬ X : A(‫)ݔ‬ > 0}.
Definition 2.2:[5] Let A and B be any two fuzzy sets in X. Then we define A ∨ B : X → [0,1] as
follows:
(A ∨ B)(‫)ݔ‬ = max {A(‫,)ݔ‬ B(‫.})ݔ‬ Also, we define A ∧ B : X → [0,1] as follows: (A ∧ B)(‫)ݔ‬ = min
{A(‫,)ݔ‬ B(‫.})ݔ‬
By A ∨ B (A ∧ B), we mean the union (intersection) between two fuzzy sets A and B of X.
Definition 2.3:[4] Let A be any fuzzy set in a set X. The complement of A, is denoted by 1X − A or Ac
and defined as follows: Ac
(‫)ݔ‬ = 1 − A(‫,)ݔ‬ for each ‫∈ݔ‬ X.
Remark 2.4: From definition (2.2) and definition (2.3) , we have, if A,B ∈ ΙX
, then A ∨ B, A ∧ B and
1X − A ∈ ΙX
.
Definition 2.5:[9] A fuzzy point ‫ݔ‬ఒ in a set X is a fuzzy set defined as follows:
‫ݔ‬ఒሺ‫ݕ‬ሻ =
Where 0 < λ ≤ 1. Now, supp(‫ݔ‬ఒ) = { ‫ݕ‬ : ‫ݔ‬ఒሺ‫ݕ‬ሻ > 0}, but
λ if ‫ݕ‬ ൌ ‫ݔ‬
0 otherwise ,
1 , for ‫∈ݔ‬ A
0 , for ‫∉ݔ‬ A ( see [4] )
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
167
‫ݔ‬ఒሺ‫ݕ‬ሻ =
supp(‫ݔ‬ఒ) = ‫ݔ‬ , so the value at ‫ݔ‬ is λ, and call the point ‫ݔ‬ its support of fuzzy point ‫ݔ‬ఒ and λ is the
height of ‫ݔ‬ఒ. That is, ‫ݔ‬ఒ has the membership degree 0 for all ‫∈ݕ‬ X except one, say ‫∈ݔ‬ X.
Definition 2.6:[3] A fuzzy topology on a set X is a family T of fuzzy sets in X which satisfies the
following conditions:
(i) 0X , 1X ∈ T,
(ii) If A , B ∈ T , then A ∧ B ∈ T,
(iii) If {Ai : i ∈ J } is a family in T, then ∨ Ai ∈ T.
T is called a fuzzy topology for X and the pair ሺܺ, ܶሻ(or simply X) is a fuzzy topological space or fts
for short. Every element of T is called T - fuzzy open set (fuzzy open set, for short ). A fuzzy set is T
- fuzzy closed (or simply fuzzy closed), if its complement is fuzzy open set. As ordinary topologies,
the indiscrete fuzzy topology on X contains only 0X and 1X (i.e., φ, X) , while the discrete fuzzy
topology on X contains all fuzzy sets in X.
Example 2.7: Let X = [−1,1], and let B1 , B2 and B3 are fuzzy sets in X defined as follows:
B1(‫)ݔ‬ =
B2(‫)ݔ‬ =
B3(‫)ݔ‬ =
Let T = { 0X , B1 , B2 , B3 , B1∨B3 , 1X }, then T is a fuzzy topology on X, and ሺܺ, ܶሻ is a fts.
Example 2.8: Let X = [0,1] and T = {0X, K , 1X }. Then T is a fuzzy topology on X, where K : X →
[0,1] defined as:
K(‫)ݔ‬ = ‫ݔ‬2
, for all ‫∈ݔ‬ X, and ሺܺ, ܶሻ is a fts.
Definition 2.9:[3] Let A be any fuzzy set in a fts X. The interior of A is the union of all fuzzy open
sets contained in A, denoted by int(A). That is, int(A) = ∨{B : B is fuzzy open set , B ≤ A}.
Definition 2.10:[3] Let A be any fuzzy set in a fts X. The closure of A is the intersection of all fuzzy
closed sets containing A, denoted by cl(A). That is, cl(A) = ∧{B : B is fuzzy closed set, B ≥ A}.
λ if ‫ݕ‬ ൌ ‫ݔ‬
0 otherwise , and 0 < λ ≤ 1 . Then,
i∈ J
1 , if −1 ≤ ‫ݔ‬ < 0
0 , if 0 ≤ ‫ݔ‬ ≤ 1
,
0 , if −1 ≤ ‫ݔ‬ < 0
1 , if 0 ≤ ‫ݔ‬ ≤ 1
,
0 , if −1 ≤ ‫ݔ‬ < 0
1/5 , if 0 ≤ ‫ݔ‬ ≤ 1
.
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
168
The most important properties of the closure and interior of fuzzy sets are listed in the
following proposition.
Proposition 2.11:[12] If A is any fuzzy set in X then:
(i) A is a fuzzy open (closed) set if and only if A = int(A) ( A = cl(A)),
(ii) cl(1X − A) = 1X − int(A),
(iii) int(1X − A) = 1X − cl(A).
Proposition 2.12:[13] Let A,B be two fuzzy sets in a fts X. Then:
(i) int(A) ≤ A, int(int(A)) = int(A),
(ii) int(A) ≤ int(B), whenever A ≤ B,
(iii) int(A∧B) = int(A) ∧ int(B), int(A∨B) ≥ int(A) ∨ int(B),
(iv) A ≤ cl(A), cl(cl(A)) = cl(A),
(v) cl(A) ≤ cl(B), whenever A ≤ B,
(vi) cl(A∧B) ≤ cl(A) ∧ cl(B), cl(A∨B) = cl(A) ∨ cl(B).
Definition 2.13:[1] Let ሺܺ, ܶሻ be a fuzzy topological space. Then ܺ is called:
(i) fuzzy ܶ଴- space (‫ܶܨ‬଴- space, for short) iff for each pair of distinct fuzzy points in ܺ, there exists
an fuzzy open set in ܺ containing one and not the other .
(ii) fuzzy ܶଵ- space (‫ܶܨ‬ଵ- space, for short) iff for each pair of distinct fuzzy points ‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ,
there exists an fuzzy open sets ‫,ܩ‬ ‫ܪ‬ containing ‫ݔ‬ఒ and ‫ݕ‬ఈ respectively such that ‫ݕ‬ఈ ‫ב‬ ‫ܩ‬ and ‫ݔ‬ఒ ‫ב‬ ‫.ܪ‬
(iii) fuzzy ܶଶ- space (‫ܶܨ‬ଶ- space, for short) iff for each pair of distinct fuzzy points ‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ,
there exist disjoint fuzzy open sets ‫,ܩ‬ ‫ܪ‬ in ܺ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ and ‫ݕ‬ఈ ‫א‬ ‫.ܪ‬
(iv) fuzzy regular space iff for each fuzzy closed set ‫ܣ‬ and for each ‫ݔ‬ఒ ‫ב‬ ‫,ܣ‬ there exist disjoint fuzzy
open sets ‫,ܩ‬ ‫ܪ‬ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ and ‫ܣ‬ ൑ ‫.ܪ‬
(v) fuzzy normal space iff for each pair of disjoint fuzzy closed sets ‫ܣ‬ and ‫,ܤ‬ there exist disjoint
fuzzy open sets ‫ܩ‬ and ‫ܪ‬ such that ‫ܣ‬ ൑ ‫ܩ‬ and ‫ܤ‬ ൑ ‫.ܪ‬
3. KERNEL SET IN FUZZY TOPOLOGICAL SPACES
Definition 3.1: The intersection of all fuzzy open subsets of a fuzzy topological space ሺܺ, ܶሻ
containing ‫ܣ‬ is called the kernel of ‫ܣ‬ (briefly ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ), this means that ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ר‬ ሼ‫ܩ‬ ‫א‬ ܶ: ‫ܣ‬ ൑ ‫ܩ‬ሽ.
Definition 3.2: In a fuzzy topological space ሺܺ, ܶሻ, a set ‫ܣ‬ is said to be weakly ultra separated from
‫ܤ‬ if there exists a fuzzy open set ‫ܩ‬ such that ‫ܣ‬ ൑ ‫ܩ‬ and ‫ܩ‬ ‫ר‬ ‫ܤ‬ ൌ 0௑ or ‫ܣ‬ ‫ר‬ ݈ܿሺ‫ܤ‬ሻ ൌ 0௑.
Remark 3.3: By definition (3.2), we have the following: For every two distinct fuzzy points
‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ ‫ݕ‬ఈ: ሼ‫ݔ‬ఒሽ is not weakly ultra separated from ሼ ‫ݕ‬ఈሽ ሽ.
Definition 3.4: A fuzzy topological space ሺܺ, ܶሻ is called fuzzy ܴ଴- space (‫ܴܨ‬଴- space, for short) if
for each fuzzy open set ܷ and ‫ݔ‬ఒ ‫א‬ ܷ then ݈ܿሼ‫ݔ‬ఒሽ ൑ ܷ.
Definition 3.5: A fuzzy topological space ሺܺ, ܶሻ is called fuzzy ܴଵ- space (‫ܴܨ‬ଵ- space, for short) if
for each two distinct fuzzy points ‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ with ݈ܿሼ‫ݔ‬ఒሽ ് ݈ܿሼ ‫ݕ‬ఈሽ, there exist disjoint fuzzy
open sets ܷ, ܸ such that ݈ܿሼ‫ݔ‬ఒሽ ൑ ܷ and ݈ܿሼ‫ݔ‬ఒሽ ൑ ܸ.
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
169
Remark 3.6: Each fuzzy separation axiom is defined as the conjunction of two weaker axioms: ‫ܶܨ‬௜-
space ൌ ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬௜ିଵ- space ൌ ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬଴- space, ݅ ൌ 1,2.
Lemma 3.7: Let ሺܺ, ܶሻ be a fuzzy topological space then ‫ݕ‬ఈ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ iff ‫ݔ‬ఒ ‫א‬ ݈ܿሼ ‫ݕ‬ఈሽ, for each
‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ.
Proof: Let ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ, then there exist fuzzy open set ܸ containing ‫ݔ‬ఒ such that ‫ݕ‬ఈ ‫ב‬ ܸ. Thus,
‫ݔ‬ఒ ‫ב‬ ݈ܿሼ ‫ݕ‬ఈሽ.
The converse part can be proved in a similar way.
Theorem 3.8: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ,
‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ and ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, there exists an fuzzy open sets ܷ, ܸ such
that ‫ݔ‬ఒ ‫א‬ ܷ, ‫ݕ‬ఈ ‫ב‬ ܷ and ‫ݕ‬ఈ ‫א‬ ܸ, ‫ݔ‬ఒ ‫ב‬ ܸ. Implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ and ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ.
Conversely, let ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ and ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ, for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ. Then there exists an fuzzy
open sets ܷ, ܸ such that
‫ݔ‬ఒ ‫א‬ ܷ, ‫ݕ‬ఈ ‫ב‬ ܷ and ‫ݕ‬ఈ ‫א‬ ܸ, ‫ݔ‬ఒ ‫ב‬ ܸ. Thus, ሺܺ, ܶሻ is a ‫ܶܨ‬ଵ- space.
Theorem 3.9: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if for each ‫ݔ‬ ‫א‬ ܺ then
݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space and let ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ് ሼ‫ݔ‬ఒሽ, then ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ contains another fuzzy point
distinct from ‫ݔ‬ఒ say ‫ݕ‬ఈ.
So ‫ݕ‬ఈ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. Hence by theorem (3.8), ሺܺ, ܶሻ is not a ‫ܶܨ‬ଵ- space this is a contradiction. Thus,
݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ.
Conversely, let ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ, for each ‫ݔ‬ ‫א‬ ܺ and let ሺܺ, ܶሻ be not a ‫ܶܨ‬ଵ- space. Then, by theorem
(3.8) , ‫ݕ‬ఈ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ, implies ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ് ሼ‫ݔ‬ఒሽ, this is a contradiction. Thus, ሺܺ, ܶሻ is a ‫ܶܨ‬ଵ- space.
Definition 3.10: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ‫ݔ‬ఒ is said to be kernelled
fuzzy point of ‫ܣ‬ ൑ ܺ (Briefly ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫))ܣ‬ if and only if for each ‫ܩ‬ fuzzy closed set contains ‫ݔ‬ఒ
then ‫ܩ‬ ‫ר‬ ‫ܣ‬ ് 0௑.
Definition 3.11: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ‫ݔ‬ఒ is said to be boundary
kernelled fuzzy point of ‫ܣ‬ (Briefly ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬௕ௗሺ‫))ܣ‬ if and only if for each fuzzy closed set ‫ܩ‬
contains ‫ݔ‬ఒ then ‫ܩ‬ ‫ר‬ ‫ܣ‬ ് 0௑ and ‫ܩ‬ ‫ר‬ ‫ܣ‬௖
് 0௑.
Definition 3.12: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ‫ݔ‬ఒ is said to be derived
kernelled fuzzy point of ‫ܣ‬ (Briefly ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬ௗ௥ሺ‫))ܣ‬ if and only if for each ‫ܩ‬ fuzzy closed set contains
‫ݔ‬ఒ then ‫ܣ‬ ‫ר‬ ‫ܩ‬ ോ ሼ‫ݔ‬ఒሽ ് 0௑.
Definition 3.13: By definition (3.10), we have the following: For every two distinct fuzzy points
‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ = ሼ‫ݕ‬ఈ: ‫ݔ‬ఒ ‫א‬ ‫ܩ‬௬ഀ
, ‫ܩ‬௬ഀ
௖
‫א‬ ܶሽ.
Theorem 3.14: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ. Then ‫ݔ‬ఒ is a kernelled fuzzy
point of ሼ ‫ݕ‬ఈሽ if and only if ‫ݕ‬ఈ is an adherent fuzzy point of ሼ‫ݔ‬ఒሽ.
Proof: Let ‫ݔ‬ఒ be a kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ. Then for every fuzzy closed set ‫ܩ‬ such that ‫ݔ‬ఒ ‫א‬
‫ܩ‬ implies ‫ݕ‬ఈ ‫א‬ ‫,ܩ‬ then
‫ݕ‬ఈ ‫א‬ ‫ר‬ ሼ‫:ܩ‬ ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ሽ, this means ‫ݕ‬ఈ ‫א‬ ݈ܿሼ‫ݔ‬ఒሽ. Thus ‫ݕ‬ఈ is an adherent fuzzy point of ሼ‫ݔ‬ఒሽ.
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
170
Conversely, let ‫ݕ‬ఈ be an adherent fuzzy point of ሼ‫ݔ‬ఒሽ. Then for every fuzzy open set ܷ such that
‫ݕ‬ఈ ‫א‬ ܷ implies ‫ݔ‬ఒ ‫א‬ ܷ, then ‫ݔ‬ఒ ‫א‬ ‫ר‬ ሼܷ: ‫ݕ‬ఈ ‫א‬ ܷሽ, this means ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ. Thus, ‫ݔ‬ఒ is a kernelled
fuzzy point of ሼ ‫ݕ‬ఈሽ.
Theorem 3.15: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ܣ‬ ൑ ܺ and let ݇‫ݎ‬ௗ௥ሺ‫)ܣ‬ be the set of all
kernelled derived fuzzy points of ‫,ܣ‬ then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫.)ܣ‬
Proof: Let ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫)ܣ‬ and if ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ, then for every fuzzy closed set ‫ܩ‬ intersects ‫ܣ‬ (in a
fuzzy point different from ‫ݔ‬ఒ). Therefore, ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. Hence, ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, it follows
that ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. To demonstrate the reverse inclusion, we consider ‫ݔ‬ఒ be a fuzzy point
of ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. If ‫ݔ‬ఒ ‫א‬ ‫,ܣ‬ then ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ൑ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ. Suppose that ‫ݔ‬ఒ ‫ב‬ ‫.ܣ‬ Since ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, then for
every fuzzy closed set ‫ܩ‬ containing ‫ݔ‬ఒ implies ‫ܩ‬ ‫ר‬ ‫ܣ‬ ് 0௑, this means ‫ܣ‬ ‫ר‬ ‫ܩ‬ ോ ሼ‫ݔ‬ఒሽ ് 0௑.
Then, ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ, so that ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ.
Theorem 3.16: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ܣ‬ ൑ ܺ and let ݇‫ݎ‬௕ௗሺ‫)ܣ‬ be the set of all
kernelled boundary fuzzy points of ‫,ܣ‬ then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫.)ܣ‬
Proof: Let ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ and if ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ , then for every fuzzy closed set ‫ܩ‬ intersects ‫,ܣ‬
therefore, ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. Hence, ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, it follows that ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. To
demonstrate the reverse inclusion, we consider ‫ݔ‬ఒ be a fuzzy point of ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. If ‫ݔ‬ఒ ‫א‬ ‫,ܣ‬ then
‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. Suppose that ‫ݔ‬ఒ ‫ב‬ ‫,ܣ‬ implies ‫ݔ‬ఒ ‫א‬ ‫ܣ‬௖
. Since ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ,
then for every fuzzy closed set ‫ܩ‬ containing ‫ݔ‬ఒ implies ‫ܩ‬ ‫ר‬ ‫ܣ‬ ≠ 0௑ and ‫ܩ‬ ‫ר‬ ‫ܣ‬௖
് 0௑. Then ‫ݔ‬ఒ ‫א‬
݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ, so that ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. Hence, ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൑ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. Thus, ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ.
Corollary 3.17: Every interior fuzzy point is a kernelled fuzzy point.
Proof: Since ݅݊‫ݐ‬ሺ‫ܣ‬ሻ ൑ ‫ܣ‬ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. Thus, every interior fuzzy point is a kernelled fuzzy point.
Theorem 3.18: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ܣ‬ is a subset of ܺ. Then ‫ܣ‬ is a fuzzy
open set if and only if every ‫ݔ‬ఒ kernelled fuzzy point of ‫ܣ‬ is an interior fuzzy point of ‫.ܣ‬
Proof: Let ‫ܣ‬ be a fuzzy open set, then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ൌ ݅݊‫ݐ‬ሺ‫ܣ‬ሻ, implies every kernelled fuzzy point is
an interior fuzzy point.
Conversely, let every ‫ݔ‬ఒ kernelled fuzzy point of ‫ܣ‬ is an interior fuzzy point of ‫.ܣ‬ Then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൑
݅݊‫ݐ‬ሺ‫ܣ‬ሻ. Hence, ݅݊‫ݐ‬ሺ‫ܣ‬ሻ ൑ ‫ܣ‬ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, implies ݅݊‫ݐ‬ሺ‫ܣ‬ሻ = ‫ܣ‬ = ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. Thus ‫ܣ‬ is a fuzzy open set.
Corollary 3.19: A subset ‫ܣ‬ of ܺ is a fuzzy open set if and only if for each ‫ݔ‬ఒ kernelled fuzzy point
then ‫ݔ‬ఒ ‫ב‬ ݈ܿሺ‫ܣ‬௖ሻ.
Proof: By theorem (3.18).
Theorem 3.20: A subset ‫ܣ‬ of ܺ is a fuzzy closed set if and only if ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑.
Proof: Let ‫ܣ‬ is a fuzzy closed set. Then ‫ܣ‬௖
is a fuzzy open set, implies ‫ܣ‬௖
ൌ ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ by theorem
(3.18). Hence ‫ܣ‬ = ݈ܿሺ‫ܣ‬ሻ. Thus ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑.
Conversely, let ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑, then for each ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ, implies ‫ݔ‬ఒ ‫ב‬ ݈ܿሺ‫ܣ‬ሻ, implies
‫ݔ‬ఒ ‫א‬ ݁‫ݐݔ‬ሺ‫ܣ‬ሻ. Therefore ‫ݔ‬ఒ ‫א‬ ݅݊‫ݐ‬ሺ‫ܣ‬௖
ሻ. Hence by theorem (3.18), ‫ܣ‬௖
is a fuzzy open set. Thus ‫ܣ‬ is a
fuzzy closed set.
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
171
Theorem 3.21: A fuzzy topological space ሺܺ, ܶሻ is ‫ܴܨ‬଴- space if and only if every adherent fuzzy
point of ሼ‫ݔ‬ఒሽ is a kernelled fuzzy point of ሼ‫ݔ‬ఒሽ.
Proof: Let ሺܺ, ܶሻ be an ‫ܴܨ‬଴- space. Then, for each ‫ݔ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ = ݈ܿሼ‫ݔ‬ఒሽ by lemma (3.7). Thus,
every adherent fuzzy point of ሼ‫ݔ‬ఒሽ is a kernelled fuzzy point of ሼ‫ݔ‬ఒሽ.
Conversely, let every adherent fuzzy point of ሼ‫ݔ‬ఒሽ is a kernelled fuzzy point of ሼ‫ݔ‬ఒሽ and let ܷ ൑
ܺ and ‫ݔ‬ఒ ‫א‬ ܷ. Then ݈ܿሼ‫ݔ‬ఒሽ ൑ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ for each ‫ݔ‬ ‫א‬ ܺ. Since ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ‫ר‬ ሼܷ: ܷ ‫א‬ ܶ, ‫ݔ‬ఒ ‫א‬ ܷሽ,
implies ݈ܿሼ‫ݔ‬ఒሽ ൑ ܷ, for each ܷ fuzzy open set contains ‫ݔ‬ఒ. Thus, ሺܺ, ܶሻ is an ‫ܴܨ‬଴- space.
Theorem 3.22: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬଴- space if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ,
either ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ or ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ.
Proof: Let ሺܺ, ܶሻ be an ‫ܶܨ‬଴- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ there exists a fuzzy open set ܷ such
that ‫ݔ‬ఒ ‫א‬ ܷ, ‫ݕ‬ఈ ‫ב‬ ܷ (say), implies ‫ݕ‬ఈ ‫א‬ ܷ௖
. Hence ܷ௖
is a fuzzy closed, then ‫ݕ‬ఈ is not kernelled
fuzzy point of ሼ‫ݔ‬ఒሽ. Thus either ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ or ‫ݕ‬ఈ is not kernelled fuzzy
point of ሼ‫ݔ‬ఒሽ.
Conversely, let for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, either ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ or ‫ݕ‬ఈ is not
kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Then there exist a fuzzy closed set ‫ܩ‬ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ , ‫ܩ‬ ‫ר‬ ሼ ‫ݕ‬ఈሽ ൌ 0௑
or ‫ݕ‬ఈ ‫א‬ ‫ܩ‬ , ‫ܩ‬ ‫ר‬ ሼ‫ݔ‬ఒሽ ൌ 0௑, implies ‫ݔ‬ఒ ‫ב‬ ‫ܩ‬௖
, ‫ݕ‬ఈ ‫א‬ ‫ܩ‬௖
or ‫ݔ‬ఒ ‫א‬ ‫ܩ‬௖
, ‫ݕ‬ఈ ‫ב‬ ‫ܩ‬௖
. Hence ‫ܩ‬௖
is a fuzzy
open set. Thus, ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space.
Theorem 3.23: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑, for each
‫ݔ‬ ‫א‬ ܺ.
Proof: Let ሺܺ, ܶሻ be an ‫ܶܨ‬ଵ- space. Then for each ‫ݔ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ = ሼ‫ݔ‬ఒሽ by theorem (3.9). Since
݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ = ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ െ ሼ‫ݔ‬ఒሽ. Thus, ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑.
Conversely, let ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑. By theorem (3.14), ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ ‫ש‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ, implies ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ
ሼ‫ݔ‬ఒሽ. Hence, by theorem (3.9), ሺܺ, ܶሻ is a ‫ܶܨ‬ଵ- space.
Theorem 3.24: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ,
‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ and ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ there exist fuzzy open sets ܷ, ܸ such
that ‫ݔ‬ఒ ‫א‬ ܷ, ‫ݕ‬ఈ ‫ב‬ ܷ and ‫ݕ‬ఈ ‫א‬ ܸ, ‫ݔ‬ఒ ‫ב‬ ܸ, implies ‫ݔ‬ఒ ‫א‬ ܸ௖
, ሼ ‫ݕ‬ఈሽ ‫ר‬ ܸ௖
ൌ 0௑ and ‫ݕ‬ఈ ‫א‬ ܷ௖
, ሼ‫ݔ‬ఒሽ ‫ר‬
ܷ௖
ൌ 0௑. Hence ܷ௖
and ܸ௖
are fuzzy closed sets. Thus ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ
and ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ.
Conversely, let for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ and ‫ݕ‬ఈ is not kernelled
fuzzy point of ሼ‫ݔ‬ఒሽ. Then, there exist fuzzy closed sets ‫ܩ‬ଵ, ‫ܩ‬ଶ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ଵ , ‫ܩ‬ଵ ‫ר‬ ሼ ‫ݕ‬ఈሽ ൌ 0௑
and ‫ݕ‬ఈ ‫א‬ ‫ܩ‬ଶ, ‫ܩ‬ଶ ‫ר‬ ሼ‫ݔ‬ఒሽ ൌ 0௑, implies ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ଶ
௖
, ‫ݕ‬ఈ ‫ב‬ ‫ܩ‬ଶ
௖
and ‫ݕ‬ఈ ‫א‬ ‫ܩ‬ଵ
௖
, ‫ݔ‬ఒ ‫ב‬ ‫ܩ‬ଵ
௖
. Hence ‫ܩ‬ଵ
௖
and ‫ܩ‬ଶ
௖
are fuzzy open sets. Thus, ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space.
4. kr- Spaces in Fuzzy Topological Spaces:
Definition 4.1: A fuzzy topological space ሺܺ, ܶሻ is said to be a ݇‫-ݎ‬ space if and only if for each
subset ‫ܣ‬ of ܺ then, ݇݁‫ݎ‬ሺ‫ܣ‬ሻ is a fuzzy open set.
Definition 4.2: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is called fuzzy ܶ௞- space (‫ܶܨ‬௞- space, for short)
if and only if for each ‫ݔ‬ఒ ‫א‬ ܺ, then ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set.
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
172
Theorem 4.3: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬ଵ-space is a ‫ܶܨ‬௞- space.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space. Then, for each ‫ݔ‬ఒ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ by theorem (3.9).
As ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ െ ሼ‫ݔ‬ఒሽ, implies ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑. Thus, ሺܺ, ܶሻ is a ‫ܶܨ‬௞- space.
Theorem 4.4: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬௞- space is a ‫ܶܨ‬଴- space.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬௞- space and let ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ. Then, ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set, therefore,
there exist two cases:
(i) ‫ݕ‬ఈ ‫א‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set. Since ‫ݔ‬ఒ ‫ב‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space.
(ii) ‫ݕ‬ఈ ‫ב‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ, implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. But ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ is a fuzzy open set. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬଴-
space.
Definition 4.5: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is said to be fuzzy ܶ௅- space (‫ܶܨ‬௅- space, for
short) if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ is degenerated (empty or singleton
fuzzy set).
Theorem 4.6: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬ଵ- space is ‫ܶܨ‬௅- space.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ and ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ሼ ‫ݕ‬ఈሽ
by theorem (3.9), implies ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬௅- space.
Theorem 4.7: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬௅- space is a ‫ܶܨ‬଴- space.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬௅- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ is degenerated
(empty or singleton fuzzy set). Therefore there exist three cases:
(i) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑, implies ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space.
(ii) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ሼ‫ݔ‬ఒሽ ‫ݎ݋‬ ሼ ‫ݕ‬ఈሽ, implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ሽ or ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ implies ሺܺ, ܶሻ is a
‫ܶܨ‬଴- space.
(iii) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ൛‫ݖ‬ఉൟ , ‫ݖ‬ ് ‫ݔ‬ ് ‫ݕ‬ , ‫ݖ‬ ‫א‬ ܺ, implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ and ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ ሼ ‫ݕ‬ఈሽ,
implies ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space.
Definition 4.8: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is said to be a fuzzy ܶே- space (‫ܶܨ‬ே- space, for
short) if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ ሼ ‫ݕ‬ఈሽ is empty or {‫ݔ‬ఒ} or { ‫ݕ‬ఈ}.
Theorem 4.9: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬ଵ- space is ‫ܶܨ‬ே- space.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ே- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ and ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ሼ ‫ݕ‬ఈሽ
by theorem (3.9), implies ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬ே- space.
Theorem 4.10: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬ே- space is a ‫ܶܨ‬଴- space.
Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ே- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ is degenerated
(empty or singleton fuzzy set). Therefore there exist two cases:
(i) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑ , implies ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space.
(ii) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ሼ‫ݔ‬ఒሽ or ሼ ‫ݕ‬ఈሽ, implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ or ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ, implies ሺܺ, ܶሻ is a
‫ܶܨ‬଴- space.
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
173
Theorem 4.11: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଶ- space iff for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, then
݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑.
Proof: Let a fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଶ- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ there exist
disjoint fuzzy open sets ܷ, ܸ such that ‫ݔ‬ఒ ‫א‬ ܷ, and ‫ݕ‬ఈ ‫א‬ ܸ. Hence ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൑ ܷ and ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൑ ܸ.
Thus ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ = 0௑.
Conversely, let for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑. Since ሺܺ, ܶሻ is a fuzzy topological
݇‫-ݎ‬ space, this means kernel is a fuzzy open set. Thus ሺܺ, ܶሻ is ‫ܶܨ‬ଶ- space.
Theorem 4.12: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is fuzzy regular space iff for each ‫ܩ‬ fuzzy
closed set and ‫ݔ‬ఒ ‫ב‬ ‫,ܩ‬ then ݇݁‫ݎ‬ሺ‫ܩ‬ሻ ‫ר‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ 0௑.
Proof: By the same way of proof of theorem (4.11).
Theorem 4.13: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is fuzzy normal space iff for each disjoint fuzzy
closed sets ‫,ܩ‬ ‫,ܪ‬ then ݇݁‫ݎ‬ሺ‫ܩ‬ሻ ‫ר‬ ݇݁‫ݎ‬ሺ‫ܪ‬ሻ ൌ 0௑.
Proof: By the same way of proof of theorem (4.11).
Theorem 4.14: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space iff it is ‫ܴܨ‬଴- space and ‫ܶܨ‬௞-
space.
Proof: It follows from theorem (4.3) and remark (3.6).
Theorem 4.15: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space iff it is ‫ܴܨ‬଴- space and ‫ܶܨ‬௅-
space.
Proof: It follows from theorem (4.6) and remark (3.6).
Theorem 4.16: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if it is ‫ܴܨ‬଴- space and
‫ܶܨ‬ே- space.
Proof: It follows from theorem (4.9) and remark (3.6).
Theorem 4.17: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬௜- space if and only if it is ‫ܴܨ‬௜ିଵ- space
and ‫ܶܨ‬௞-space, ݅ ൌ 1,2.
Proof: It follows from theorem (4.3) and remark (3.6).
Theorem 4.18: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬௜- space if and only if it is ‫ܴܨ‬௜ିଵ- space
and ‫ܶܨ‬௅-space, ݅ ൌ 1,2.
Proof: It follows from theorem (4.6) and remark (3.6).
Theorem 4.19: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬௜- space if and only if it is ‫ܴܨ‬௜ିଵ- space
and ‫ܶܨ‬ே-space, ݅ ൌ 1,2.
Proof: It follows from theorem (4.9) and remark (3.6).
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME
174
Remark 4.20: The relation between fuzzy separation axioms can be representing as a matrix.
Therefore, the element ܽ௜௝ refers to this relation. As the following matrix representation shows:
Matrix Representation
The relation between fuzzy separation axioms in fuzzy topological ݇‫-ݎ‬ spaces
REFERENCES
1. A. S. Mashour, E, E. Kerre and M. H. Ghanim, “Separation axioms, sub spaces and sums in
fuzzy topology”, J. Math Anal., 102(1984), 189-202.
2. B. Sikn, “On fuzzy FC - Compactness”, Comm. Korean Math. Soc.13 (1998), 137-150.
3. C. L. Chang, “Fuzzy topological spaces”, J. Math. Anal. Appl. 24(1968), 182-190.
4. G. J. Klir, U. S. Clair, B. Yuan, “ Fuzzy set theory ”, foundations and applications, 1997.
5. G. Balasubramanian, “Fuzzy β - open sets and fuzzy β - separation axioms”, Kybernetika
35(1999), 215-223.
6. G. Balasubramania and P. Sundaram, “On some generalizations of fuzzy continuous
functions”, Fuzzy sets and Systems, 86(1) (1997), 93-100.
7. H. Maki et al., “Generalized closed sets in fuzzy topological space, I, Meetings on
Topological spaces Theory and its Applications”, (1998), 23-36.
8. L. A. Zadeh, “Fuzzy sets”, Information, and control, 8 (1965), 338 -353.
9. M. Sarkar, “On fuzzy topological spaces”, J. Math. Anal. Appl. 79 (1981), 384-394.
10. M. Alimohammady and M. Roohi, “On Fuzzy φψ - Continuous Multifunction”, J. App. Math.
sto. Anal. (2006), 1-7.
11. O. Berde Ozbaki, “On generalized fuzzy strongly semiclosed sets in fuzzy topological
spaces”, Int. J. Math. Sci. 30 (11) (2002), 651- 657.
12. R. H. Warren, “Neighborhoods, bases and continuity in fuzzy topological spaces”, The
Rocky Mountain Journal of Mathematics, Vol. 8, No.3, (1977), 459-470.
13. X. Tang, “Spatial object modeling in fuzzy topological spaces with application to lands cover
change in china”, Ph. D. Dissertation, ITC Dissertation No.108, Univ. of Twente, The Nether
lands, (2004).
14. Gunwanti S. Mahajan and Kanchan S. Bhagat, “Medical Image Segmentation using
Enhanced K-Means and Kernelized Fuzzy C- Means”, International Journal of Electronics
and Communication Engineering &Technology (IJECET), Volume 4, Issue 6, 2013,
pp. 62 - 70, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472.
and ‫ܶܨ‬଴ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܴܨ‬଴ ‫ܴܨ‬ଵ ‫ܶܨ‬௞ ‫ܶܨ‬௅ ‫ܶܨ‬ே
‫ܶܨ‬଴ ‫ܶܨ‬଴ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬௞ ‫ܶܨ‬௅ ‫ܶܨ‬ே
‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ
‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ
‫ܴܨ‬଴ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܴܨ‬଴ ‫ܴܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ
‫ܴܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܴܨ‬ଵ ‫ܴܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ
‫ܶܨ‬௞ ‫ܶܨ‬௞ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬௞ ‫ܶܨ‬௅ ‫ܶܨ‬଴
‫ܶܨ‬௅ ‫ܶܨ‬௅ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬௅ ‫ܶܨ‬௅ ‫ܶܨ‬଴
‫ܶܨ‬ே ‫ܶܨ‬ே ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬଴ ‫ܶܨ‬଴ ‫ܶܨ‬ே

Más contenido relacionado

La actualidad más candente

On Some New Continuous Mappings in Intuitionistic Fuzzy Topological Spaces
On Some New Continuous Mappings in Intuitionistic Fuzzy Topological SpacesOn Some New Continuous Mappings in Intuitionistic Fuzzy Topological Spaces
On Some New Continuous Mappings in Intuitionistic Fuzzy Topological SpacesEditor IJCATR
 
Fuzzy Homogeneous Bitopological Spaces
Fuzzy  Homogeneous Bitopological Spaces Fuzzy  Homogeneous Bitopological Spaces
Fuzzy Homogeneous Bitopological Spaces IJECEIAES
 
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological Spaces
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological Spaces(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological Spaces
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
 
Report sem 1
Report sem 1Report sem 1
Report sem 1jobishvd
 
(C f)- weak contraction in cone metric spaces
(C f)- weak contraction in cone metric spaces(C f)- weak contraction in cone metric spaces
(C f)- weak contraction in cone metric spacesAlexander Decker
 
Fuzzy Bitopological Ideals Spaces
Fuzzy Bitopological Ideals SpacesFuzzy Bitopological Ideals Spaces
Fuzzy Bitopological Ideals SpacesIOSR Journals
 
ON ANNIHILATOR OF INTUITIONISTIC FUZZY SUBSETS OF MODULES
ON ANNIHILATOR OF INTUITIONISTIC FUZZY SUBSETS OF MODULESON ANNIHILATOR OF INTUITIONISTIC FUZZY SUBSETS OF MODULES
ON ANNIHILATOR OF INTUITIONISTIC FUZZY SUBSETS OF MODULEScsandit
 
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Cristiano Longo
 
On Some New Contra Continuous and Contra Open Mappings in Intuitionistic Fuzz...
On Some New Contra Continuous and Contra Open Mappings in Intuitionistic Fuzz...On Some New Contra Continuous and Contra Open Mappings in Intuitionistic Fuzz...
On Some New Contra Continuous and Contra Open Mappings in Intuitionistic Fuzz...Editor IJCATR
 
Polycycles and their elementary decompositions
Polycycles and their elementary decompositionsPolycycles and their elementary decompositions
Polycycles and their elementary decompositionsMathieu Dutour Sikiric
 
IRJET- W-R0 Space in Minimal G-Closed Sets of Type1
IRJET- W-R0 Space in Minimal G-Closed Sets of Type1IRJET- W-R0 Space in Minimal G-Closed Sets of Type1
IRJET- W-R0 Space in Minimal G-Closed Sets of Type1IRJET Journal
 
On fixed point theorems in fuzzy metric spaces in integral type
On fixed point theorems in fuzzy metric spaces in integral typeOn fixed point theorems in fuzzy metric spaces in integral type
On fixed point theorems in fuzzy metric spaces in integral typeAlexander Decker
 
Db31463471
Db31463471Db31463471
Db31463471IJMER
 
Interval valued intuitionistic fuzzy homomorphism of bf algebras
Interval valued intuitionistic fuzzy homomorphism of bf algebrasInterval valued intuitionistic fuzzy homomorphism of bf algebras
Interval valued intuitionistic fuzzy homomorphism of bf algebrasAlexander Decker
 
Some New Operations on Fuzzy Soft Sets
Some New Operations on Fuzzy Soft SetsSome New Operations on Fuzzy Soft Sets
Some New Operations on Fuzzy Soft SetsIJMER
 
g∗S-closed sets in topological spaces
g∗S-closed sets in topological spacesg∗S-closed sets in topological spaces
g∗S-closed sets in topological spacesIJMER
 
On fuzzy compact open topology
On fuzzy compact open topologyOn fuzzy compact open topology
On fuzzy compact open topologybkrawat08
 

La actualidad más candente (19)

On Some New Continuous Mappings in Intuitionistic Fuzzy Topological Spaces
On Some New Continuous Mappings in Intuitionistic Fuzzy Topological SpacesOn Some New Continuous Mappings in Intuitionistic Fuzzy Topological Spaces
On Some New Continuous Mappings in Intuitionistic Fuzzy Topological Spaces
 
Fuzzy Homogeneous Bitopological Spaces
Fuzzy  Homogeneous Bitopological Spaces Fuzzy  Homogeneous Bitopological Spaces
Fuzzy Homogeneous Bitopological Spaces
 
1 s2.0-s1574035804702117-main
1 s2.0-s1574035804702117-main1 s2.0-s1574035804702117-main
1 s2.0-s1574035804702117-main
 
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological Spaces
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological Spaces(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological Spaces
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological Spaces
 
Ow3426032607
Ow3426032607Ow3426032607
Ow3426032607
 
Report sem 1
Report sem 1Report sem 1
Report sem 1
 
(C f)- weak contraction in cone metric spaces
(C f)- weak contraction in cone metric spaces(C f)- weak contraction in cone metric spaces
(C f)- weak contraction in cone metric spaces
 
Fuzzy Bitopological Ideals Spaces
Fuzzy Bitopological Ideals SpacesFuzzy Bitopological Ideals Spaces
Fuzzy Bitopological Ideals Spaces
 
ON ANNIHILATOR OF INTUITIONISTIC FUZZY SUBSETS OF MODULES
ON ANNIHILATOR OF INTUITIONISTIC FUZZY SUBSETS OF MODULESON ANNIHILATOR OF INTUITIONISTIC FUZZY SUBSETS OF MODULES
ON ANNIHILATOR OF INTUITIONISTIC FUZZY SUBSETS OF MODULES
 
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
 
On Some New Contra Continuous and Contra Open Mappings in Intuitionistic Fuzz...
On Some New Contra Continuous and Contra Open Mappings in Intuitionistic Fuzz...On Some New Contra Continuous and Contra Open Mappings in Intuitionistic Fuzz...
On Some New Contra Continuous and Contra Open Mappings in Intuitionistic Fuzz...
 
Polycycles and their elementary decompositions
Polycycles and their elementary decompositionsPolycycles and their elementary decompositions
Polycycles and their elementary decompositions
 
IRJET- W-R0 Space in Minimal G-Closed Sets of Type1
IRJET- W-R0 Space in Minimal G-Closed Sets of Type1IRJET- W-R0 Space in Minimal G-Closed Sets of Type1
IRJET- W-R0 Space in Minimal G-Closed Sets of Type1
 
On fixed point theorems in fuzzy metric spaces in integral type
On fixed point theorems in fuzzy metric spaces in integral typeOn fixed point theorems in fuzzy metric spaces in integral type
On fixed point theorems in fuzzy metric spaces in integral type
 
Db31463471
Db31463471Db31463471
Db31463471
 
Interval valued intuitionistic fuzzy homomorphism of bf algebras
Interval valued intuitionistic fuzzy homomorphism of bf algebrasInterval valued intuitionistic fuzzy homomorphism of bf algebras
Interval valued intuitionistic fuzzy homomorphism of bf algebras
 
Some New Operations on Fuzzy Soft Sets
Some New Operations on Fuzzy Soft SetsSome New Operations on Fuzzy Soft Sets
Some New Operations on Fuzzy Soft Sets
 
g∗S-closed sets in topological spaces
g∗S-closed sets in topological spacesg∗S-closed sets in topological spaces
g∗S-closed sets in topological spaces
 
On fuzzy compact open topology
On fuzzy compact open topologyOn fuzzy compact open topology
On fuzzy compact open topology
 

Similar a 50120140504018

H214148
H214148H214148
H214148irjes
 
On almost generalized semi continuous
On almost    generalized semi continuousOn almost    generalized semi continuous
On almost generalized semi continuousAlexander Decker
 
11.on almost generalized semi continuous
11.on almost    generalized semi continuous11.on almost    generalized semi continuous
11.on almost generalized semi continuousAlexander Decker
 
An approach to Fuzzy clustering of the iris petals by using Ac-means
An approach to Fuzzy clustering of the iris petals by using Ac-meansAn approach to Fuzzy clustering of the iris petals by using Ac-means
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
 
Residual Quotient and Annihilator of Intuitionistic Fuzzy Sets of Ring and Mo...
Residual Quotient and Annihilator of Intuitionistic Fuzzy Sets of Ring and Mo...Residual Quotient and Annihilator of Intuitionistic Fuzzy Sets of Ring and Mo...
Residual Quotient and Annihilator of Intuitionistic Fuzzy Sets of Ring and Mo...AIRCC Publishing Corporation
 
𝒈 ∗S-closed sets in topological spaces
𝒈 ∗S-closed sets in topological spaces𝒈 ∗S-closed sets in topological spaces
𝒈 ∗S-closed sets in topological spacesIJMER
 
On intuitionistic fuzzy 휷 generalized closed sets
On intuitionistic fuzzy 휷 generalized closed setsOn intuitionistic fuzzy 휷 generalized closed sets
On intuitionistic fuzzy 휷 generalized closed setsijceronline
 
Some Results on Fuzzy Supra Topological Spaces
Some Results on Fuzzy Supra Topological SpacesSome Results on Fuzzy Supra Topological Spaces
Some Results on Fuzzy Supra Topological SpacesIJERA Editor
 
On intuitionistic fuzzy 휷 generalized closed sets
On intuitionistic fuzzy 휷 generalized closed setsOn intuitionistic fuzzy 휷 generalized closed sets
On intuitionistic fuzzy 휷 generalized closed setsijceronline
 
Notions via β*-open sets in topological spaces
Notions via β*-open sets in topological spacesNotions via β*-open sets in topological spaces
Notions via β*-open sets in topological spacesIOSR Journals
 
On Zα-Open Sets and Decompositions of Continuity
On Zα-Open Sets and Decompositions of ContinuityOn Zα-Open Sets and Decompositions of Continuity
On Zα-Open Sets and Decompositions of ContinuityIJERA Editor
 
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -Continuity
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -ContinuityIntuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -Continuity
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -ContinuityWaqas Tariq
 
α -ANTI FUZZY NEW IDEAL OF PUALGEBRA
α -ANTI FUZZY NEW IDEAL OF PUALGEBRAα -ANTI FUZZY NEW IDEAL OF PUALGEBRA
α -ANTI FUZZY NEW IDEAL OF PUALGEBRAWireilla
 
a - ANTI FUZZY NEW IDEAL OF PUALGEBRA
a - ANTI FUZZY NEW IDEAL OF PUALGEBRAa - ANTI FUZZY NEW IDEAL OF PUALGEBRA
a - ANTI FUZZY NEW IDEAL OF PUALGEBRAijfls
 
On Review of the Cluster Point of a Set in a Topological Space
On Review of the Cluster Point of a Set in a Topological SpaceOn Review of the Cluster Point of a Set in a Topological Space
On Review of the Cluster Point of a Set in a Topological SpaceBRNSS Publication Hub
 
applied mathematics methods.pdf
applied mathematics methods.pdfapplied mathematics methods.pdf
applied mathematics methods.pdfCyprianObota
 

Similar a 50120140504018 (20)

Au4103292297
Au4103292297Au4103292297
Au4103292297
 
H214148
H214148H214148
H214148
 
On almost generalized semi continuous
On almost    generalized semi continuousOn almost    generalized semi continuous
On almost generalized semi continuous
 
11.on almost generalized semi continuous
11.on almost    generalized semi continuous11.on almost    generalized semi continuous
11.on almost generalized semi continuous
 
Fuzzy algebra
Fuzzy algebra Fuzzy algebra
Fuzzy algebra
 
An approach to Fuzzy clustering of the iris petals by using Ac-means
An approach to Fuzzy clustering of the iris petals by using Ac-meansAn approach to Fuzzy clustering of the iris petals by using Ac-means
An approach to Fuzzy clustering of the iris petals by using Ac-means
 
Residual Quotient and Annihilator of Intuitionistic Fuzzy Sets of Ring and Mo...
Residual Quotient and Annihilator of Intuitionistic Fuzzy Sets of Ring and Mo...Residual Quotient and Annihilator of Intuitionistic Fuzzy Sets of Ring and Mo...
Residual Quotient and Annihilator of Intuitionistic Fuzzy Sets of Ring and Mo...
 
𝒈 ∗S-closed sets in topological spaces
𝒈 ∗S-closed sets in topological spaces𝒈 ∗S-closed sets in topological spaces
𝒈 ∗S-closed sets in topological spaces
 
On intuitionistic fuzzy 휷 generalized closed sets
On intuitionistic fuzzy 휷 generalized closed setsOn intuitionistic fuzzy 휷 generalized closed sets
On intuitionistic fuzzy 휷 generalized closed sets
 
Some Results on Fuzzy Supra Topological Spaces
Some Results on Fuzzy Supra Topological SpacesSome Results on Fuzzy Supra Topological Spaces
Some Results on Fuzzy Supra Topological Spaces
 
On intuitionistic fuzzy 휷 generalized closed sets
On intuitionistic fuzzy 휷 generalized closed setsOn intuitionistic fuzzy 휷 generalized closed sets
On intuitionistic fuzzy 휷 generalized closed sets
 
Notions via β*-open sets in topological spaces
Notions via β*-open sets in topological spacesNotions via β*-open sets in topological spaces
Notions via β*-open sets in topological spaces
 
On Zα-Open Sets and Decompositions of Continuity
On Zα-Open Sets and Decompositions of ContinuityOn Zα-Open Sets and Decompositions of Continuity
On Zα-Open Sets and Decompositions of Continuity
 
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -Continuity
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -ContinuityIntuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -Continuity
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -Continuity
 
α -ANTI FUZZY NEW IDEAL OF PUALGEBRA
α -ANTI FUZZY NEW IDEAL OF PUALGEBRAα -ANTI FUZZY NEW IDEAL OF PUALGEBRA
α -ANTI FUZZY NEW IDEAL OF PUALGEBRA
 
a - ANTI FUZZY NEW IDEAL OF PUALGEBRA
a - ANTI FUZZY NEW IDEAL OF PUALGEBRAa - ANTI FUZZY NEW IDEAL OF PUALGEBRA
a - ANTI FUZZY NEW IDEAL OF PUALGEBRA
 
01_AJMS_211_19_REV.pdf
01_AJMS_211_19_REV.pdf01_AJMS_211_19_REV.pdf
01_AJMS_211_19_REV.pdf
 
01_AJMS_211_19_REV.pdf
01_AJMS_211_19_REV.pdf01_AJMS_211_19_REV.pdf
01_AJMS_211_19_REV.pdf
 
On Review of the Cluster Point of a Set in a Topological Space
On Review of the Cluster Point of a Set in a Topological SpaceOn Review of the Cluster Point of a Set in a Topological Space
On Review of the Cluster Point of a Set in a Topological Space
 
applied mathematics methods.pdf
applied mathematics methods.pdfapplied mathematics methods.pdf
applied mathematics methods.pdf
 

Más de IAEME Publication

IAEME_Publication_Call_for_Paper_September_2022.pdf
IAEME_Publication_Call_for_Paper_September_2022.pdfIAEME_Publication_Call_for_Paper_September_2022.pdf
IAEME_Publication_Call_for_Paper_September_2022.pdfIAEME Publication
 
MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...
MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...
MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...IAEME Publication
 
A STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURS
A STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURSA STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURS
A STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURSIAEME Publication
 
BROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURS
BROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURSBROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURS
BROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURSIAEME Publication
 
DETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONS
DETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONSDETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONS
DETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONSIAEME Publication
 
ANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONS
ANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONSANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONS
ANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONSIAEME Publication
 
VOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINO
VOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINOVOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINO
VOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINOIAEME Publication
 
IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...
IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...
IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...IAEME Publication
 
VISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMY
VISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMYVISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMY
VISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMYIAEME Publication
 
A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...
A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...
A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...IAEME Publication
 
GANDHI ON NON-VIOLENT POLICE
GANDHI ON NON-VIOLENT POLICEGANDHI ON NON-VIOLENT POLICE
GANDHI ON NON-VIOLENT POLICEIAEME Publication
 
A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...
A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...
A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...IAEME Publication
 
ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...
ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...
ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...IAEME Publication
 
INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...
INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...
INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...IAEME Publication
 
A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...
A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...
A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...IAEME Publication
 
EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...
EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...
EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...IAEME Publication
 
ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...
ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...
ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...IAEME Publication
 
OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...
OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...
OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...IAEME Publication
 
APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...
APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...
APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...IAEME Publication
 
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENT
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENTA MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENT
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENTIAEME Publication
 

Más de IAEME Publication (20)

IAEME_Publication_Call_for_Paper_September_2022.pdf
IAEME_Publication_Call_for_Paper_September_2022.pdfIAEME_Publication_Call_for_Paper_September_2022.pdf
IAEME_Publication_Call_for_Paper_September_2022.pdf
 
MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...
MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...
MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...
 
A STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURS
A STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURSA STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURS
A STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURS
 
BROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURS
BROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURSBROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURS
BROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURS
 
DETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONS
DETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONSDETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONS
DETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONS
 
ANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONS
ANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONSANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONS
ANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONS
 
VOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINO
VOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINOVOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINO
VOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINO
 
IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...
IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...
IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...
 
VISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMY
VISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMYVISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMY
VISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMY
 
A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...
A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...
A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...
 
GANDHI ON NON-VIOLENT POLICE
GANDHI ON NON-VIOLENT POLICEGANDHI ON NON-VIOLENT POLICE
GANDHI ON NON-VIOLENT POLICE
 
A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...
A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...
A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...
 
ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...
ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...
ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...
 
INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...
INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...
INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...
 
A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...
A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...
A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...
 
EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...
EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...
EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...
 
ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...
ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...
ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...
 
OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...
OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...
OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...
 
APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...
APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...
APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...
 
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENT
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENTA MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENT
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENT
 

Último

NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopBachir Benyammi
 
UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6DianaGray10
 
OpenShift Commons Paris - Choose Your Own Observability Adventure
OpenShift Commons Paris - Choose Your Own Observability AdventureOpenShift Commons Paris - Choose Your Own Observability Adventure
OpenShift Commons Paris - Choose Your Own Observability AdventureEric D. Schabell
 
Empowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintEmpowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintMahmoud Rabie
 
Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1DianaGray10
 
9 Steps For Building Winning Founding Team
9 Steps For Building Winning Founding Team9 Steps For Building Winning Founding Team
9 Steps For Building Winning Founding TeamAdam Moalla
 
Videogame localization & technology_ how to enhance the power of translation.pdf
Videogame localization & technology_ how to enhance the power of translation.pdfVideogame localization & technology_ how to enhance the power of translation.pdf
Videogame localization & technology_ how to enhance the power of translation.pdfinfogdgmi
 
Linked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesLinked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesDavid Newbury
 
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostKubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostMatt Ray
 
How Accurate are Carbon Emissions Projections?
How Accurate are Carbon Emissions Projections?How Accurate are Carbon Emissions Projections?
How Accurate are Carbon Emissions Projections?IES VE
 
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1DianaGray10
 
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationUsing IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationIES VE
 
COMPUTER 10 Lesson 8 - Building a Website
COMPUTER 10 Lesson 8 - Building a WebsiteCOMPUTER 10 Lesson 8 - Building a Website
COMPUTER 10 Lesson 8 - Building a Websitedgelyza
 
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsIgniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsSafe Software
 
AI You Can Trust - Ensuring Success with Data Integrity Webinar
AI You Can Trust - Ensuring Success with Data Integrity WebinarAI You Can Trust - Ensuring Success with Data Integrity Webinar
AI You Can Trust - Ensuring Success with Data Integrity WebinarPrecisely
 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPathCommunity
 
Nanopower In Semiconductor Industry.pdf
Nanopower  In Semiconductor Industry.pdfNanopower  In Semiconductor Industry.pdf
Nanopower In Semiconductor Industry.pdfPedro Manuel
 
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UbiTrack UK
 
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfIaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfDaniel Santiago Silva Capera
 

Último (20)

NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 Workshop
 
UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6
 
OpenShift Commons Paris - Choose Your Own Observability Adventure
OpenShift Commons Paris - Choose Your Own Observability AdventureOpenShift Commons Paris - Choose Your Own Observability Adventure
OpenShift Commons Paris - Choose Your Own Observability Adventure
 
Empowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintEmpowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership Blueprint
 
Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1
 
9 Steps For Building Winning Founding Team
9 Steps For Building Winning Founding Team9 Steps For Building Winning Founding Team
9 Steps For Building Winning Founding Team
 
20150722 - AGV
20150722 - AGV20150722 - AGV
20150722 - AGV
 
Videogame localization & technology_ how to enhance the power of translation.pdf
Videogame localization & technology_ how to enhance the power of translation.pdfVideogame localization & technology_ how to enhance the power of translation.pdf
Videogame localization & technology_ how to enhance the power of translation.pdf
 
Linked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesLinked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond Ontologies
 
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostKubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
 
How Accurate are Carbon Emissions Projections?
How Accurate are Carbon Emissions Projections?How Accurate are Carbon Emissions Projections?
How Accurate are Carbon Emissions Projections?
 
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
 
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationUsing IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
 
COMPUTER 10 Lesson 8 - Building a Website
COMPUTER 10 Lesson 8 - Building a WebsiteCOMPUTER 10 Lesson 8 - Building a Website
COMPUTER 10 Lesson 8 - Building a Website
 
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsIgniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
 
AI You Can Trust - Ensuring Success with Data Integrity Webinar
AI You Can Trust - Ensuring Success with Data Integrity WebinarAI You Can Trust - Ensuring Success with Data Integrity Webinar
AI You Can Trust - Ensuring Success with Data Integrity Webinar
 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation Developers
 
Nanopower In Semiconductor Industry.pdf
Nanopower  In Semiconductor Industry.pdfNanopower  In Semiconductor Industry.pdf
Nanopower In Semiconductor Industry.pdf
 
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
 
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfIaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
 

50120140504018

  • 1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 165 NEW CHARACTERIZATION OF KERNEL SET IN FUZZY TOPOLOGICAL SPACES Qays Hatem Imran Al-Muthanna University, College of Education, Department of Mathematics, Al-Muthanna, Iraq Basim Mohammed Melgat Foundation of Technical Education, Technical Institute of Al-Mussaib, Babylon, Iraq ABSTRACT In this paper we introduce a kernelled fuzzy point, boundary kernelled fuzzy point and derived kernelled fuzzy point of a subset A of X, and using these notions to define kernel set of fuzzy topological spaces. Also we introduce fuzzy topological ݇‫-ݎ‬ space. The investigation enables us to present some new fuzzy separation axioms between ‫ܶܨ‬଴ and ‫ܶܨ‬ଵ- spaces. Keywords: Fuzzy Topological Space, Kernelled Fuzzy Point, Boundary Kernelled Fuzzy Point, Derived Kernelled Fuzzy Point, Kernel Set, Weak Fuzzy Separation Axioms, ‫ܴܨ‬௜- space, ݅ ൌ 0,1. 1. INTRODUCTION The concept of fuzzy set and fuzzy set operations were first introduced by L. A. Zadeh in 1965 [8]. After Zadeh 's introduction of fuzzy sets, Chang [3] defined and studied the notion of fuzzy topological space in 1968. In 1997, fuzzy generalized closed set (Fg - closed set) was introduced by G. Balasubramania and P. Sundaram [6]. In 1998, the notion of Fgs - closed set was defined and investigated by H. Maki el al [7]. In 2002, O. Bedre Ozbakir [11], defined the concept of fuzzy generalized strongly closed set. In 1984, fuzzy separation axioms have been introduced and investigated by A. S. Mashour and others [1]. In this paper we introduce a new characterization of kernel set through our definition kernelled fuzzy point, boundary kernelled fuzzy point and derived kernelled fuzzy point. By these notions, we obtain that the kernel of a set in fuzzy topological space ሺܺ, ܶሻ is a union of the set itself with the set of all boundary kernelled fuzzy points. In addition, it is a union of the set itself with the set of all derived kernelled fuzzy points and we give some result of ‫ܴܨ‬଴- space by using these notions. Also in this paper we introduce fuzzy topological ݇‫-ݎ‬ space iff kernel of a subset A of X is INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2014): 8.5328 (Calculated by GISI) www.jifactor.com IJCET © I A E M E
  • 2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 166 an fuzzy open set. Via this kind of fuzzy topological space, we give a new characterization of fuzzy separation axioms lying between ‫ܶܨ‬଴ and ‫ܶܨ‬ଵ- spaces. 2. PRELIMINARIES Fuzzy sets theory, introduced by Lotfi. A. Zada in 1965 [8], is the extension of classical set theory by allowing the membership of elements to range from 0 to 1. Let X be the universe of a classical set of objects. Membership in a classical subset A of X is often viewed as a characteristic function µA from X into {0, 1}, where µA(‫)ݔ‬ = for any ‫∈ݔ‬ X. {0,1} is called a valuation set (see [13]). If the valuation set is allowed to be the real interval [0,1], A is called a fuzzy set in X. µA(‫)ݔ‬ (or simply A(‫))ݔ‬ is the membership value (or degree of membership) of ‫ݔ‬ in A. Clearly, A is a subset of X that has no sharp boundary. A fuzzy set A in X can be represented by the set of pairs: A ={( ‫ݔ‬ , A(‫,))ݔ‬ ‫∈ݔ‬ X}. Let A : X → [0,1] be a fuzzy set. If A(‫)ݔ‬ = 1, for each ‫∈ݔ‬ X, we denote it by 1X and if A(‫)ݔ‬ = 0, for each ‫∈ݔ‬ X, we denote it by 0X . That is, by 0X and 1X , we mean the constant fuzzy sets taking the values 0 and 1 on X, respectively [2]. Let Ι =[0,1]. The set of all fuzzy sets in X, denoted by ΙX [10]. Definition 2.1:[4] Let A be a fuzzy set of a set X. The support of A is the elements ‫ݔ‬ whose membership value is greater than 0, i.e., supp(A) = { ‫∈ݔ‬ X : A(‫)ݔ‬ > 0}. Definition 2.2:[5] Let A and B be any two fuzzy sets in X. Then we define A ∨ B : X → [0,1] as follows: (A ∨ B)(‫)ݔ‬ = max {A(‫,)ݔ‬ B(‫.})ݔ‬ Also, we define A ∧ B : X → [0,1] as follows: (A ∧ B)(‫)ݔ‬ = min {A(‫,)ݔ‬ B(‫.})ݔ‬ By A ∨ B (A ∧ B), we mean the union (intersection) between two fuzzy sets A and B of X. Definition 2.3:[4] Let A be any fuzzy set in a set X. The complement of A, is denoted by 1X − A or Ac and defined as follows: Ac (‫)ݔ‬ = 1 − A(‫,)ݔ‬ for each ‫∈ݔ‬ X. Remark 2.4: From definition (2.2) and definition (2.3) , we have, if A,B ∈ ΙX , then A ∨ B, A ∧ B and 1X − A ∈ ΙX . Definition 2.5:[9] A fuzzy point ‫ݔ‬ఒ in a set X is a fuzzy set defined as follows: ‫ݔ‬ఒሺ‫ݕ‬ሻ = Where 0 < λ ≤ 1. Now, supp(‫ݔ‬ఒ) = { ‫ݕ‬ : ‫ݔ‬ఒሺ‫ݕ‬ሻ > 0}, but λ if ‫ݕ‬ ൌ ‫ݔ‬ 0 otherwise , 1 , for ‫∈ݔ‬ A 0 , for ‫∉ݔ‬ A ( see [4] )
  • 3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 167 ‫ݔ‬ఒሺ‫ݕ‬ሻ = supp(‫ݔ‬ఒ) = ‫ݔ‬ , so the value at ‫ݔ‬ is λ, and call the point ‫ݔ‬ its support of fuzzy point ‫ݔ‬ఒ and λ is the height of ‫ݔ‬ఒ. That is, ‫ݔ‬ఒ has the membership degree 0 for all ‫∈ݕ‬ X except one, say ‫∈ݔ‬ X. Definition 2.6:[3] A fuzzy topology on a set X is a family T of fuzzy sets in X which satisfies the following conditions: (i) 0X , 1X ∈ T, (ii) If A , B ∈ T , then A ∧ B ∈ T, (iii) If {Ai : i ∈ J } is a family in T, then ∨ Ai ∈ T. T is called a fuzzy topology for X and the pair ሺܺ, ܶሻ(or simply X) is a fuzzy topological space or fts for short. Every element of T is called T - fuzzy open set (fuzzy open set, for short ). A fuzzy set is T - fuzzy closed (or simply fuzzy closed), if its complement is fuzzy open set. As ordinary topologies, the indiscrete fuzzy topology on X contains only 0X and 1X (i.e., φ, X) , while the discrete fuzzy topology on X contains all fuzzy sets in X. Example 2.7: Let X = [−1,1], and let B1 , B2 and B3 are fuzzy sets in X defined as follows: B1(‫)ݔ‬ = B2(‫)ݔ‬ = B3(‫)ݔ‬ = Let T = { 0X , B1 , B2 , B3 , B1∨B3 , 1X }, then T is a fuzzy topology on X, and ሺܺ, ܶሻ is a fts. Example 2.8: Let X = [0,1] and T = {0X, K , 1X }. Then T is a fuzzy topology on X, where K : X → [0,1] defined as: K(‫)ݔ‬ = ‫ݔ‬2 , for all ‫∈ݔ‬ X, and ሺܺ, ܶሻ is a fts. Definition 2.9:[3] Let A be any fuzzy set in a fts X. The interior of A is the union of all fuzzy open sets contained in A, denoted by int(A). That is, int(A) = ∨{B : B is fuzzy open set , B ≤ A}. Definition 2.10:[3] Let A be any fuzzy set in a fts X. The closure of A is the intersection of all fuzzy closed sets containing A, denoted by cl(A). That is, cl(A) = ∧{B : B is fuzzy closed set, B ≥ A}. λ if ‫ݕ‬ ൌ ‫ݔ‬ 0 otherwise , and 0 < λ ≤ 1 . Then, i∈ J 1 , if −1 ≤ ‫ݔ‬ < 0 0 , if 0 ≤ ‫ݔ‬ ≤ 1 , 0 , if −1 ≤ ‫ݔ‬ < 0 1 , if 0 ≤ ‫ݔ‬ ≤ 1 , 0 , if −1 ≤ ‫ݔ‬ < 0 1/5 , if 0 ≤ ‫ݔ‬ ≤ 1 .
  • 4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 168 The most important properties of the closure and interior of fuzzy sets are listed in the following proposition. Proposition 2.11:[12] If A is any fuzzy set in X then: (i) A is a fuzzy open (closed) set if and only if A = int(A) ( A = cl(A)), (ii) cl(1X − A) = 1X − int(A), (iii) int(1X − A) = 1X − cl(A). Proposition 2.12:[13] Let A,B be two fuzzy sets in a fts X. Then: (i) int(A) ≤ A, int(int(A)) = int(A), (ii) int(A) ≤ int(B), whenever A ≤ B, (iii) int(A∧B) = int(A) ∧ int(B), int(A∨B) ≥ int(A) ∨ int(B), (iv) A ≤ cl(A), cl(cl(A)) = cl(A), (v) cl(A) ≤ cl(B), whenever A ≤ B, (vi) cl(A∧B) ≤ cl(A) ∧ cl(B), cl(A∨B) = cl(A) ∨ cl(B). Definition 2.13:[1] Let ሺܺ, ܶሻ be a fuzzy topological space. Then ܺ is called: (i) fuzzy ܶ଴- space (‫ܶܨ‬଴- space, for short) iff for each pair of distinct fuzzy points in ܺ, there exists an fuzzy open set in ܺ containing one and not the other . (ii) fuzzy ܶଵ- space (‫ܶܨ‬ଵ- space, for short) iff for each pair of distinct fuzzy points ‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ, there exists an fuzzy open sets ‫,ܩ‬ ‫ܪ‬ containing ‫ݔ‬ఒ and ‫ݕ‬ఈ respectively such that ‫ݕ‬ఈ ‫ב‬ ‫ܩ‬ and ‫ݔ‬ఒ ‫ב‬ ‫.ܪ‬ (iii) fuzzy ܶଶ- space (‫ܶܨ‬ଶ- space, for short) iff for each pair of distinct fuzzy points ‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ, there exist disjoint fuzzy open sets ‫,ܩ‬ ‫ܪ‬ in ܺ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ and ‫ݕ‬ఈ ‫א‬ ‫.ܪ‬ (iv) fuzzy regular space iff for each fuzzy closed set ‫ܣ‬ and for each ‫ݔ‬ఒ ‫ב‬ ‫,ܣ‬ there exist disjoint fuzzy open sets ‫,ܩ‬ ‫ܪ‬ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ and ‫ܣ‬ ൑ ‫.ܪ‬ (v) fuzzy normal space iff for each pair of disjoint fuzzy closed sets ‫ܣ‬ and ‫,ܤ‬ there exist disjoint fuzzy open sets ‫ܩ‬ and ‫ܪ‬ such that ‫ܣ‬ ൑ ‫ܩ‬ and ‫ܤ‬ ൑ ‫.ܪ‬ 3. KERNEL SET IN FUZZY TOPOLOGICAL SPACES Definition 3.1: The intersection of all fuzzy open subsets of a fuzzy topological space ሺܺ, ܶሻ containing ‫ܣ‬ is called the kernel of ‫ܣ‬ (briefly ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ), this means that ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ר‬ ሼ‫ܩ‬ ‫א‬ ܶ: ‫ܣ‬ ൑ ‫ܩ‬ሽ. Definition 3.2: In a fuzzy topological space ሺܺ, ܶሻ, a set ‫ܣ‬ is said to be weakly ultra separated from ‫ܤ‬ if there exists a fuzzy open set ‫ܩ‬ such that ‫ܣ‬ ൑ ‫ܩ‬ and ‫ܩ‬ ‫ר‬ ‫ܤ‬ ൌ 0௑ or ‫ܣ‬ ‫ר‬ ݈ܿሺ‫ܤ‬ሻ ൌ 0௑. Remark 3.3: By definition (3.2), we have the following: For every two distinct fuzzy points ‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ ‫ݕ‬ఈ: ሼ‫ݔ‬ఒሽ is not weakly ultra separated from ሼ ‫ݕ‬ఈሽ ሽ. Definition 3.4: A fuzzy topological space ሺܺ, ܶሻ is called fuzzy ܴ଴- space (‫ܴܨ‬଴- space, for short) if for each fuzzy open set ܷ and ‫ݔ‬ఒ ‫א‬ ܷ then ݈ܿሼ‫ݔ‬ఒሽ ൑ ܷ. Definition 3.5: A fuzzy topological space ሺܺ, ܶሻ is called fuzzy ܴଵ- space (‫ܴܨ‬ଵ- space, for short) if for each two distinct fuzzy points ‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ with ݈ܿሼ‫ݔ‬ఒሽ ് ݈ܿሼ ‫ݕ‬ఈሽ, there exist disjoint fuzzy open sets ܷ, ܸ such that ݈ܿሼ‫ݔ‬ఒሽ ൑ ܷ and ݈ܿሼ‫ݔ‬ఒሽ ൑ ܸ.
  • 5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 169 Remark 3.6: Each fuzzy separation axiom is defined as the conjunction of two weaker axioms: ‫ܶܨ‬௜- space ൌ ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬௜ିଵ- space ൌ ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬଴- space, ݅ ൌ 1,2. Lemma 3.7: Let ሺܺ, ܶሻ be a fuzzy topological space then ‫ݕ‬ఈ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ iff ‫ݔ‬ఒ ‫א‬ ݈ܿሼ ‫ݕ‬ఈሽ, for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ. Proof: Let ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ, then there exist fuzzy open set ܸ containing ‫ݔ‬ఒ such that ‫ݕ‬ఈ ‫ב‬ ܸ. Thus, ‫ݔ‬ఒ ‫ב‬ ݈ܿሼ ‫ݕ‬ఈሽ. The converse part can be proved in a similar way. Theorem 3.8: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ and ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, there exists an fuzzy open sets ܷ, ܸ such that ‫ݔ‬ఒ ‫א‬ ܷ, ‫ݕ‬ఈ ‫ב‬ ܷ and ‫ݕ‬ఈ ‫א‬ ܸ, ‫ݔ‬ఒ ‫ב‬ ܸ. Implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ and ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ. Conversely, let ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ and ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ, for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ. Then there exists an fuzzy open sets ܷ, ܸ such that ‫ݔ‬ఒ ‫א‬ ܷ, ‫ݕ‬ఈ ‫ב‬ ܷ and ‫ݕ‬ఈ ‫א‬ ܸ, ‫ݔ‬ఒ ‫ב‬ ܸ. Thus, ሺܺ, ܶሻ is a ‫ܶܨ‬ଵ- space. Theorem 3.9: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if for each ‫ݔ‬ ‫א‬ ܺ then ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space and let ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ് ሼ‫ݔ‬ఒሽ, then ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ contains another fuzzy point distinct from ‫ݔ‬ఒ say ‫ݕ‬ఈ. So ‫ݕ‬ఈ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. Hence by theorem (3.8), ሺܺ, ܶሻ is not a ‫ܶܨ‬ଵ- space this is a contradiction. Thus, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ. Conversely, let ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ, for each ‫ݔ‬ ‫א‬ ܺ and let ሺܺ, ܶሻ be not a ‫ܶܨ‬ଵ- space. Then, by theorem (3.8) , ‫ݕ‬ఈ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ, implies ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ് ሼ‫ݔ‬ఒሽ, this is a contradiction. Thus, ሺܺ, ܶሻ is a ‫ܶܨ‬ଵ- space. Definition 3.10: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ‫ݔ‬ఒ is said to be kernelled fuzzy point of ‫ܣ‬ ൑ ܺ (Briefly ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫))ܣ‬ if and only if for each ‫ܩ‬ fuzzy closed set contains ‫ݔ‬ఒ then ‫ܩ‬ ‫ר‬ ‫ܣ‬ ് 0௑. Definition 3.11: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ‫ݔ‬ఒ is said to be boundary kernelled fuzzy point of ‫ܣ‬ (Briefly ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬௕ௗሺ‫))ܣ‬ if and only if for each fuzzy closed set ‫ܩ‬ contains ‫ݔ‬ఒ then ‫ܩ‬ ‫ר‬ ‫ܣ‬ ് 0௑ and ‫ܩ‬ ‫ר‬ ‫ܣ‬௖ ് 0௑. Definition 3.12: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ‫ݔ‬ఒ is said to be derived kernelled fuzzy point of ‫ܣ‬ (Briefly ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬ௗ௥ሺ‫))ܣ‬ if and only if for each ‫ܩ‬ fuzzy closed set contains ‫ݔ‬ఒ then ‫ܣ‬ ‫ר‬ ‫ܩ‬ ോ ሼ‫ݔ‬ఒሽ ് 0௑. Definition 3.13: By definition (3.10), we have the following: For every two distinct fuzzy points ‫ݔ‬ఒ and ‫ݕ‬ఈ of ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ = ሼ‫ݕ‬ఈ: ‫ݔ‬ఒ ‫א‬ ‫ܩ‬௬ഀ , ‫ܩ‬௬ഀ ௖ ‫א‬ ܶሽ. Theorem 3.14: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ. Then ‫ݔ‬ఒ is a kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ if and only if ‫ݕ‬ఈ is an adherent fuzzy point of ሼ‫ݔ‬ఒሽ. Proof: Let ‫ݔ‬ఒ be a kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ. Then for every fuzzy closed set ‫ܩ‬ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ implies ‫ݕ‬ఈ ‫א‬ ‫,ܩ‬ then ‫ݕ‬ఈ ‫א‬ ‫ר‬ ሼ‫:ܩ‬ ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ሽ, this means ‫ݕ‬ఈ ‫א‬ ݈ܿሼ‫ݔ‬ఒሽ. Thus ‫ݕ‬ఈ is an adherent fuzzy point of ሼ‫ݔ‬ఒሽ.
  • 6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 170 Conversely, let ‫ݕ‬ఈ be an adherent fuzzy point of ሼ‫ݔ‬ఒሽ. Then for every fuzzy open set ܷ such that ‫ݕ‬ఈ ‫א‬ ܷ implies ‫ݔ‬ఒ ‫א‬ ܷ, then ‫ݔ‬ఒ ‫א‬ ‫ר‬ ሼܷ: ‫ݕ‬ఈ ‫א‬ ܷሽ, this means ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ. Thus, ‫ݔ‬ఒ is a kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ. Theorem 3.15: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ܣ‬ ൑ ܺ and let ݇‫ݎ‬ௗ௥ሺ‫)ܣ‬ be the set of all kernelled derived fuzzy points of ‫,ܣ‬ then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫.)ܣ‬ Proof: Let ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫)ܣ‬ and if ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ, then for every fuzzy closed set ‫ܩ‬ intersects ‫ܣ‬ (in a fuzzy point different from ‫ݔ‬ఒ). Therefore, ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. Hence, ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, it follows that ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. To demonstrate the reverse inclusion, we consider ‫ݔ‬ఒ be a fuzzy point of ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. If ‫ݔ‬ఒ ‫א‬ ‫,ܣ‬ then ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ൑ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ. Suppose that ‫ݔ‬ఒ ‫ב‬ ‫.ܣ‬ Since ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, then for every fuzzy closed set ‫ܩ‬ containing ‫ݔ‬ఒ implies ‫ܩ‬ ‫ר‬ ‫ܣ‬ ് 0௑, this means ‫ܣ‬ ‫ר‬ ‫ܩ‬ ോ ሼ‫ݔ‬ఒሽ ് 0௑. Then, ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ, so that ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ. Theorem 3.16: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ܣ‬ ൑ ܺ and let ݇‫ݎ‬௕ௗሺ‫)ܣ‬ be the set of all kernelled boundary fuzzy points of ‫,ܣ‬ then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫.)ܣ‬ Proof: Let ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ and if ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ , then for every fuzzy closed set ‫ܩ‬ intersects ‫,ܣ‬ therefore, ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. Hence, ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, it follows that ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. To demonstrate the reverse inclusion, we consider ‫ݔ‬ఒ be a fuzzy point of ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. If ‫ݔ‬ఒ ‫א‬ ‫,ܣ‬ then ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. Suppose that ‫ݔ‬ఒ ‫ב‬ ‫,ܣ‬ implies ‫ݔ‬ఒ ‫א‬ ‫ܣ‬௖ . Since ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, then for every fuzzy closed set ‫ܩ‬ containing ‫ݔ‬ఒ implies ‫ܩ‬ ‫ר‬ ‫ܣ‬ ≠ 0௑ and ‫ܩ‬ ‫ר‬ ‫ܣ‬௖ ് 0௑. Then ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ, so that ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. Hence, ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൑ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. Thus, ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ. Corollary 3.17: Every interior fuzzy point is a kernelled fuzzy point. Proof: Since ݅݊‫ݐ‬ሺ‫ܣ‬ሻ ൑ ‫ܣ‬ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. Thus, every interior fuzzy point is a kernelled fuzzy point. Theorem 3.18: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ܣ‬ is a subset of ܺ. Then ‫ܣ‬ is a fuzzy open set if and only if every ‫ݔ‬ఒ kernelled fuzzy point of ‫ܣ‬ is an interior fuzzy point of ‫.ܣ‬ Proof: Let ‫ܣ‬ be a fuzzy open set, then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ൌ ݅݊‫ݐ‬ሺ‫ܣ‬ሻ, implies every kernelled fuzzy point is an interior fuzzy point. Conversely, let every ‫ݔ‬ఒ kernelled fuzzy point of ‫ܣ‬ is an interior fuzzy point of ‫.ܣ‬ Then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൑ ݅݊‫ݐ‬ሺ‫ܣ‬ሻ. Hence, ݅݊‫ݐ‬ሺ‫ܣ‬ሻ ൑ ‫ܣ‬ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, implies ݅݊‫ݐ‬ሺ‫ܣ‬ሻ = ‫ܣ‬ = ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. Thus ‫ܣ‬ is a fuzzy open set. Corollary 3.19: A subset ‫ܣ‬ of ܺ is a fuzzy open set if and only if for each ‫ݔ‬ఒ kernelled fuzzy point then ‫ݔ‬ఒ ‫ב‬ ݈ܿሺ‫ܣ‬௖ሻ. Proof: By theorem (3.18). Theorem 3.20: A subset ‫ܣ‬ of ܺ is a fuzzy closed set if and only if ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑. Proof: Let ‫ܣ‬ is a fuzzy closed set. Then ‫ܣ‬௖ is a fuzzy open set, implies ‫ܣ‬௖ ൌ ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ by theorem (3.18). Hence ‫ܣ‬ = ݈ܿሺ‫ܣ‬ሻ. Thus ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑. Conversely, let ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑, then for each ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ, implies ‫ݔ‬ఒ ‫ב‬ ݈ܿሺ‫ܣ‬ሻ, implies ‫ݔ‬ఒ ‫א‬ ݁‫ݐݔ‬ሺ‫ܣ‬ሻ. Therefore ‫ݔ‬ఒ ‫א‬ ݅݊‫ݐ‬ሺ‫ܣ‬௖ ሻ. Hence by theorem (3.18), ‫ܣ‬௖ is a fuzzy open set. Thus ‫ܣ‬ is a fuzzy closed set.
  • 7. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 171 Theorem 3.21: A fuzzy topological space ሺܺ, ܶሻ is ‫ܴܨ‬଴- space if and only if every adherent fuzzy point of ሼ‫ݔ‬ఒሽ is a kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Proof: Let ሺܺ, ܶሻ be an ‫ܴܨ‬଴- space. Then, for each ‫ݔ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ = ݈ܿሼ‫ݔ‬ఒሽ by lemma (3.7). Thus, every adherent fuzzy point of ሼ‫ݔ‬ఒሽ is a kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Conversely, let every adherent fuzzy point of ሼ‫ݔ‬ఒሽ is a kernelled fuzzy point of ሼ‫ݔ‬ఒሽ and let ܷ ൑ ܺ and ‫ݔ‬ఒ ‫א‬ ܷ. Then ݈ܿሼ‫ݔ‬ఒሽ ൑ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ for each ‫ݔ‬ ‫א‬ ܺ. Since ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ‫ר‬ ሼܷ: ܷ ‫א‬ ܶ, ‫ݔ‬ఒ ‫א‬ ܷሽ, implies ݈ܿሼ‫ݔ‬ఒሽ ൑ ܷ, for each ܷ fuzzy open set contains ‫ݔ‬ఒ. Thus, ሺܺ, ܶሻ is an ‫ܴܨ‬଴- space. Theorem 3.22: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬଴- space if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, either ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ or ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Proof: Let ሺܺ, ܶሻ be an ‫ܶܨ‬଴- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ there exists a fuzzy open set ܷ such that ‫ݔ‬ఒ ‫א‬ ܷ, ‫ݕ‬ఈ ‫ב‬ ܷ (say), implies ‫ݕ‬ఈ ‫א‬ ܷ௖ . Hence ܷ௖ is a fuzzy closed, then ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Thus either ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ or ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Conversely, let for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, either ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ or ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Then there exist a fuzzy closed set ‫ܩ‬ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ , ‫ܩ‬ ‫ר‬ ሼ ‫ݕ‬ఈሽ ൌ 0௑ or ‫ݕ‬ఈ ‫א‬ ‫ܩ‬ , ‫ܩ‬ ‫ר‬ ሼ‫ݔ‬ఒሽ ൌ 0௑, implies ‫ݔ‬ఒ ‫ב‬ ‫ܩ‬௖ , ‫ݕ‬ఈ ‫א‬ ‫ܩ‬௖ or ‫ݔ‬ఒ ‫א‬ ‫ܩ‬௖ , ‫ݕ‬ఈ ‫ב‬ ‫ܩ‬௖ . Hence ‫ܩ‬௖ is a fuzzy open set. Thus, ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space. Theorem 3.23: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑, for each ‫ݔ‬ ‫א‬ ܺ. Proof: Let ሺܺ, ܶሻ be an ‫ܶܨ‬ଵ- space. Then for each ‫ݔ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ = ሼ‫ݔ‬ఒሽ by theorem (3.9). Since ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ = ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ െ ሼ‫ݔ‬ఒሽ. Thus, ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑. Conversely, let ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑. By theorem (3.14), ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ ‫ש‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ, implies ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ. Hence, by theorem (3.9), ሺܺ, ܶሻ is a ‫ܶܨ‬ଵ- space. Theorem 3.24: A fuzzy topological space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ and ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ there exist fuzzy open sets ܷ, ܸ such that ‫ݔ‬ఒ ‫א‬ ܷ, ‫ݕ‬ఈ ‫ב‬ ܷ and ‫ݕ‬ఈ ‫א‬ ܸ, ‫ݔ‬ఒ ‫ב‬ ܸ, implies ‫ݔ‬ఒ ‫א‬ ܸ௖ , ሼ ‫ݕ‬ఈሽ ‫ר‬ ܸ௖ ൌ 0௑ and ‫ݕ‬ఈ ‫א‬ ܷ௖ , ሼ‫ݔ‬ఒሽ ‫ר‬ ܷ௖ ൌ 0௑. Hence ܷ௖ and ܸ௖ are fuzzy closed sets. Thus ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ and ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Conversely, let for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ‫ݔ‬ఒ is not kernelled fuzzy point of ሼ ‫ݕ‬ఈሽ and ‫ݕ‬ఈ is not kernelled fuzzy point of ሼ‫ݔ‬ఒሽ. Then, there exist fuzzy closed sets ‫ܩ‬ଵ, ‫ܩ‬ଶ such that ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ଵ , ‫ܩ‬ଵ ‫ר‬ ሼ ‫ݕ‬ఈሽ ൌ 0௑ and ‫ݕ‬ఈ ‫א‬ ‫ܩ‬ଶ, ‫ܩ‬ଶ ‫ר‬ ሼ‫ݔ‬ఒሽ ൌ 0௑, implies ‫ݔ‬ఒ ‫א‬ ‫ܩ‬ଶ ௖ , ‫ݕ‬ఈ ‫ב‬ ‫ܩ‬ଶ ௖ and ‫ݕ‬ఈ ‫א‬ ‫ܩ‬ଵ ௖ , ‫ݔ‬ఒ ‫ב‬ ‫ܩ‬ଵ ௖ . Hence ‫ܩ‬ଵ ௖ and ‫ܩ‬ଶ ௖ are fuzzy open sets. Thus, ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space. 4. kr- Spaces in Fuzzy Topological Spaces: Definition 4.1: A fuzzy topological space ሺܺ, ܶሻ is said to be a ݇‫-ݎ‬ space if and only if for each subset ‫ܣ‬ of ܺ then, ݇݁‫ݎ‬ሺ‫ܣ‬ሻ is a fuzzy open set. Definition 4.2: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is called fuzzy ܶ௞- space (‫ܶܨ‬௞- space, for short) if and only if for each ‫ݔ‬ఒ ‫א‬ ܺ, then ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set.
  • 8. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 172 Theorem 4.3: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬ଵ-space is a ‫ܶܨ‬௞- space. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space. Then, for each ‫ݔ‬ఒ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ by theorem (3.9). As ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ െ ሼ‫ݔ‬ఒሽ, implies ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑. Thus, ሺܺ, ܶሻ is a ‫ܶܨ‬௞- space. Theorem 4.4: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬௞- space is a ‫ܶܨ‬଴- space. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬௞- space and let ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ. Then, ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set, therefore, there exist two cases: (i) ‫ݕ‬ఈ ‫א‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set. Since ‫ݔ‬ఒ ‫ב‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space. (ii) ‫ݕ‬ఈ ‫ב‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ, implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. But ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ is a fuzzy open set. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space. Definition 4.5: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is said to be fuzzy ܶ௅- space (‫ܶܨ‬௅- space, for short) if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ is degenerated (empty or singleton fuzzy set). Theorem 4.6: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬ଵ- space is ‫ܶܨ‬௅- space. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ଵ- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ and ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ሼ ‫ݕ‬ఈሽ by theorem (3.9), implies ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬௅- space. Theorem 4.7: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬௅- space is a ‫ܶܨ‬଴- space. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬௅- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ is degenerated (empty or singleton fuzzy set). Therefore there exist three cases: (i) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑, implies ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space. (ii) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ሼ‫ݔ‬ఒሽ ‫ݎ݋‬ ሼ ‫ݕ‬ఈሽ, implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ሽ or ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ implies ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space. (iii) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ൛‫ݖ‬ఉൟ , ‫ݖ‬ ് ‫ݔ‬ ് ‫ݕ‬ , ‫ݖ‬ ‫א‬ ܺ, implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ and ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ ሼ ‫ݕ‬ఈሽ, implies ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space. Definition 4.8: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is said to be a fuzzy ܶே- space (‫ܶܨ‬ே- space, for short) if and only if for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ ሼ ‫ݕ‬ఈሽ is empty or {‫ݔ‬ఒ} or { ‫ݕ‬ఈ}. Theorem 4.9: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬ଵ- space is ‫ܶܨ‬ே- space. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ே- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ and ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ሼ ‫ݕ‬ఈሽ by theorem (3.9), implies ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬ே- space. Theorem 4.10: In fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ, every ‫ܶܨ‬ே- space is a ‫ܶܨ‬଴- space. Proof: Let ሺܺ, ܶሻ be a ‫ܶܨ‬ே- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ is degenerated (empty or singleton fuzzy set). Therefore there exist two cases: (i) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑ , implies ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space. (ii) ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ ሼ‫ݔ‬ఒሽ or ሼ ‫ݕ‬ఈሽ, implies ‫ݕ‬ఈ ‫ב‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ or ‫ݔ‬ఒ ‫ב‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ, implies ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space.
  • 9. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 173 Theorem 4.11: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଶ- space iff for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, then ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑. Proof: Let a fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଶ- space. Then for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ there exist disjoint fuzzy open sets ܷ, ܸ such that ‫ݔ‬ఒ ‫א‬ ܷ, and ‫ݕ‬ఈ ‫א‬ ܸ. Hence ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൑ ܷ and ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൑ ܸ. Thus ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ = 0௑. Conversely, let for each ‫ݔ‬ ് ‫ݕ‬ ‫א‬ ܺ, ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ‫ר‬ ݇݁‫ݎ‬ሼ ‫ݕ‬ఈሽ ൌ 0௑. Since ሺܺ, ܶሻ is a fuzzy topological ݇‫-ݎ‬ space, this means kernel is a fuzzy open set. Thus ሺܺ, ܶሻ is ‫ܶܨ‬ଶ- space. Theorem 4.12: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is fuzzy regular space iff for each ‫ܩ‬ fuzzy closed set and ‫ݔ‬ఒ ‫ב‬ ‫,ܩ‬ then ݇݁‫ݎ‬ሺ‫ܩ‬ሻ ‫ר‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ 0௑. Proof: By the same way of proof of theorem (4.11). Theorem 4.13: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is fuzzy normal space iff for each disjoint fuzzy closed sets ‫,ܩ‬ ‫,ܪ‬ then ݇݁‫ݎ‬ሺ‫ܩ‬ሻ ‫ר‬ ݇݁‫ݎ‬ሺ‫ܪ‬ሻ ൌ 0௑. Proof: By the same way of proof of theorem (4.11). Theorem 4.14: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space iff it is ‫ܴܨ‬଴- space and ‫ܶܨ‬௞- space. Proof: It follows from theorem (4.3) and remark (3.6). Theorem 4.15: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space iff it is ‫ܴܨ‬଴- space and ‫ܶܨ‬௅- space. Proof: It follows from theorem (4.6) and remark (3.6). Theorem 4.16: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if it is ‫ܴܨ‬଴- space and ‫ܶܨ‬ே- space. Proof: It follows from theorem (4.9) and remark (3.6). Theorem 4.17: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬௜- space if and only if it is ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬௞-space, ݅ ൌ 1,2. Proof: It follows from theorem (4.3) and remark (3.6). Theorem 4.18: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬௜- space if and only if it is ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬௅-space, ݅ ൌ 1,2. Proof: It follows from theorem (4.6) and remark (3.6). Theorem 4.19: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬௜- space if and only if it is ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬ே-space, ݅ ൌ 1,2. Proof: It follows from theorem (4.9) and remark (3.6).
  • 10. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 174 Remark 4.20: The relation between fuzzy separation axioms can be representing as a matrix. Therefore, the element ܽ௜௝ refers to this relation. As the following matrix representation shows: Matrix Representation The relation between fuzzy separation axioms in fuzzy topological ݇‫-ݎ‬ spaces REFERENCES 1. A. S. Mashour, E, E. Kerre and M. H. Ghanim, “Separation axioms, sub spaces and sums in fuzzy topology”, J. Math Anal., 102(1984), 189-202. 2. B. Sikn, “On fuzzy FC - Compactness”, Comm. Korean Math. Soc.13 (1998), 137-150. 3. C. L. Chang, “Fuzzy topological spaces”, J. Math. Anal. Appl. 24(1968), 182-190. 4. G. J. Klir, U. S. Clair, B. Yuan, “ Fuzzy set theory ”, foundations and applications, 1997. 5. G. Balasubramanian, “Fuzzy β - open sets and fuzzy β - separation axioms”, Kybernetika 35(1999), 215-223. 6. G. Balasubramania and P. Sundaram, “On some generalizations of fuzzy continuous functions”, Fuzzy sets and Systems, 86(1) (1997), 93-100. 7. H. Maki et al., “Generalized closed sets in fuzzy topological space, I, Meetings on Topological spaces Theory and its Applications”, (1998), 23-36. 8. L. A. Zadeh, “Fuzzy sets”, Information, and control, 8 (1965), 338 -353. 9. M. Sarkar, “On fuzzy topological spaces”, J. Math. Anal. Appl. 79 (1981), 384-394. 10. M. Alimohammady and M. Roohi, “On Fuzzy φψ - Continuous Multifunction”, J. App. Math. sto. Anal. (2006), 1-7. 11. O. Berde Ozbaki, “On generalized fuzzy strongly semiclosed sets in fuzzy topological spaces”, Int. J. Math. Sci. 30 (11) (2002), 651- 657. 12. R. H. Warren, “Neighborhoods, bases and continuity in fuzzy topological spaces”, The Rocky Mountain Journal of Mathematics, Vol. 8, No.3, (1977), 459-470. 13. X. Tang, “Spatial object modeling in fuzzy topological spaces with application to lands cover change in china”, Ph. D. Dissertation, ITC Dissertation No.108, Univ. of Twente, The Nether lands, (2004). 14. Gunwanti S. Mahajan and Kanchan S. Bhagat, “Medical Image Segmentation using Enhanced K-Means and Kernelized Fuzzy C- Means”, International Journal of Electronics and Communication Engineering &Technology (IJECET), Volume 4, Issue 6, 2013, pp. 62 - 70, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472. and ‫ܶܨ‬଴ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܴܨ‬଴ ‫ܴܨ‬ଵ ‫ܶܨ‬௞ ‫ܶܨ‬௅ ‫ܶܨ‬ே ‫ܶܨ‬଴ ‫ܶܨ‬଴ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬௞ ‫ܶܨ‬௅ ‫ܶܨ‬ே ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܴܨ‬଴ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܴܨ‬଴ ‫ܴܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܶܨ‬ଵ ‫ܴܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܴܨ‬ଵ ‫ܴܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬ଶ ‫ܶܨ‬௞ ‫ܶܨ‬௞ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬௞ ‫ܶܨ‬௅ ‫ܶܨ‬଴ ‫ܶܨ‬௅ ‫ܶܨ‬௅ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬௅ ‫ܶܨ‬௅ ‫ܶܨ‬଴ ‫ܶܨ‬ே ‫ܶܨ‬ே ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬ଵ ‫ܶܨ‬ଶ ‫ܶܨ‬଴ ‫ܶܨ‬଴ ‫ܶܨ‬ே