50120140504018

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
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ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 5, Issue 4, April (2014), pp. 165-174
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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] )

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‫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
.

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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 ൑ ܸ.

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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.

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.

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.

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.

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.
Theorem 4.16: A fuzzy topological ݇‫-ݎ‬ space ሺܺ, ܶሻ is ‫ܶܨ‬ଵ- space if and only if it is ‫ܴܨ‬଴- space and
‫ܶܨ‬ே- space.
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.
and ‫ܶܨ‬௅-space, ݅ ൌ 1,2.
and ‫ܶܨ‬ே-space, ݅ ൌ 1,2.

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
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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

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