p∗-Closure Operator and p∗-Regularity in Fuzzy Setting

In this paper a new type of fuzzy regularity, viz. fuzzy p∗regularity has been introduced and studied by a newly defined closure operator, viz., fuzzy p∗-closure operator. Also we have found the mutual relationship of this closure operator among other closure operators defined earlier. In p∗-regular space, p∗-closure operator is an idempotent operator. In the last section, p∗-closure operator has been characterized via p∗-convergence of a fuzzy net.


Introduction
Throughout the paper, by (X, τ ) or simply by X we mean a fuzzy topological space (fts, for short) in the sense of Chang [3].A fuzzy set [7] A is a mapping from a nonempty set X into a closed interval I = [0, 1].The support [6] of a fuzzy set A in X will be denoted by suppA and is defined by suppA = {x ∈ X : A(x) = 0}.A fuzzy point [6] with the singleton support x ∈ X and the value t (0 < t ≤ 1) at x will be denoted by x t .0 X and 1 X are the constant fuzzy sets taking values 0 and 1 in X respectively.The complement [7] of a fuzzy set A in X will be denoted by 1 X A and is defined by (1 X A)(x) = 1 − A(x), for all x ∈ X.For two fuzzy sets A and B in X, we write A ≤ B if and only if A(x) ≤ B(x), for each x ∈ X, and AqB means A is quasi-coincident (q-coincident, for short) with B [6] if A(x) + B(x) > 1, for some x ∈ X.The negation of these two statements will be denoted by A ≤ B and A qB respectively.clA and intA of a fuzzy set A in X respectively stand for the fuzzy closure [3] and fuzzy interior [3] of A in X.A fuzzy set A in X is called fuzzy α-open [2] if A ≤ intclintA.The complement of a fuzzy α-open set is called a fuzzy α-closed [2]  The union of all fuzzy preopen sets contained in a fuzzy set A is called fuzzy preinterior of A, to be denoted by pintA.
The intersection of all fuzzy preclosed sets containing a fuzzy set A is called fuzzy preclosure of A, to be denoted by pclA.Definition 2.2.A fuzzy preopen set A in an fts (X, τ ) is called a fuzzy pre-q-nbd of a fuzzy point x t , if x t qA.Lemma 2.1.For a fuzzy point x t and a fuzzy set A in an fts (X, τ ), x t ∈ pclA if and only if every fuzzy pre-q-nbd U of x t , U qA.
Proof.Let x t ∈ pclA and U be any fuzzy pre-q-nbd of x t .Then U (x U which is fuzzy preclosed in X and hence by Definition 2.1, Conversely, let for any fuzzy pre-q-nbd U of x t , U qA. Let V be any fuzzy preclosed set containing A, i.e., A ≤ V ... (1).We have to show that Lemma 2.2.For any two fuzzy preopen sets A and B in an fts X, A qB ⇒ pclA qB and A qpclB.
Proof.If possible, let pclAqB.Then there exists x ∈ X such that pclA(x) + B(x) > 1.Let pclA(x) = t.Then B(x) + t > 1 ⇒ x t qB and x t ∈ pclA.By Lemma 2.1, BqA, a contradiction.Similarly, we can prove that A qpclB.Definition 2.3.A fuzzy point x t in an fts X is called fuzzy p * -cluster point of a fuzzy set A in X if pclU qA for every fuzzy pre-q-nbd U of x t .The union of all fuzzy p * -cluster points of a fuzzy set A is called fuzzy p *closure of A, to be denoted by Note 2.1.It is clear from Definition 2.1 and Definition 2.3 that pclA ≤ [A] p , for any fuzzy set A in an fts X.The converse is not true, in general, as seen from the following example.
Consider the fuzzy point a 0.2 and the fuzzy set V defined by The following theorem shows that under which condition, the two closure operators pcl and p * coincide.
Proof.By Note 2.1, it suffices to show that [A] p ≤ pclA, for any fuzzy preopen set A in X.Let x t be a fuzzy point in X such that x t ∈ pclA.Then there exists a fuzzy pre-q-nbd V of x t such that V qA.Then V (y Thus pclV qA and consequently, x t ∈ [A] p .Hence [A] p ≤ pclA for a fuzzy preopen set A in X.
We now characterize fuzzy p * -closure operator of a fuzzy set A in an fts X.

Theorem 2.2. For any fuzzy set
Then there exists a fuzzy pre-qnbd V of x t such that pclV qA and so A ≤ 1 X pclV and 1 X pclV being fuzzy preopen set in X containing A, by our assumption, x t ∈ [1 X pclV ] p .But pclV q(1 X pclV ) and so x t ∈ [1 X pclV ] p , a contradiction.This completes the proof.
Remark 2.1.By Theorem 2.1 and Theorem 2.2, we can conclude that [A] p is fuzzy preclosed in X for a fuzzy set A in X.
Theorem 2.3.In an fts (X, τ ), the following hold: Therefore, pclV q(A B) for any fuzzy pre-q-nbd V of x t and hence Remark 2.2.In fact, the intersection of any collection of fuzzy p * -closed sets is fuzzy p * -closed.But the union of two fuzzy p * -closed sets may not be fuzzy p * -closed is clear from the following example.
Example 2.2.Let X = {a, b}, τ = {0 X , 1 X , A} where A(a) = 0.4, A(b) = 0.7.Then (X, τ ) is an fts.The collection of all fuzzy preopen sets in (X, τ ) is {0 X , 1 X , A, U } where U ≤ 1 X A. Then the collection of all fuzzy preclosed sets is {0 X , 1 X , 1 X A, 1 X U } where 1 X U ≥ A. But for any fuzzy pre-q-nbd of a 0.6 is of the form U where U ≤ 1 X A. Then pclU = U q(C D) and consequently, a 0  Then the collection of all fuzzy preclosed sets is Consider the fuzzy points a 0.6 and b 0.1 .We claim that b 0.1 ∈ [a 0.6 ] p , but a 0.6 / ∈ [b 0.1 ] p .Indeed, any fuzzy preq-nbd of b 0.1 is of the form V where V (a) > 0.5, V (b) > 0.9 and pclV = W where W (a) > 0.5, W (b) = 1 and W qa 0.6 .But D(a) = 0.41, D(b) = 0 is a fuzzy pre-q-nbd of a 0.6 and pclD = D qb 0.1 .

p * -Closure Operator: Mutual Relationship with Other Closure Operators
In this section we have established some mutual relationship of p * -closure operator with other closure operators, viz., α * -closure operator, θ-closure operator.
First We recall some definitions for ready references.

Definition 3.1 ([5]
).Let A be a fuzzy set and x t , a fuzzy point in an fts X. x t is called a fuzzy θ-cluster point of A if every closure of every fuzzy open q-nbd of x t is q-coincident with A.
The union of all fuzzy θ-cluster points of A is called fuzzy θ-closure of A, to be denoted by

Definition 3.2 ([1]
).A fuzzy point x t in an fts X is called a fuzzy α * -cluster point of a fuzzy set A in X if αclU qA for every fuzzy α-open q-nbd U of x t .The union of all fuzzy α * -cluster points of A is called fuzzy α * -closure of A, to be denoted by Proof.Let x t ∈ [A] p .Let V be any fuzzy open q-nbd of x t .Then V is fuzzy pre-q-nbd of x t also.As  Example 3.4.Let X = {a}, τ = {0 X , 1 X , A, B} where A(a) = 0.4, B(a) = 0.7.Then (X, τ ) is an fts.Then the collection of all fuzzy preopen sets is {0 X , 1 X , U, V } where U ≤ A, V ≥ B. Consider the fuzzy point a 0.4 and the fuzzy set C given by C(a) = 0.3.Then B is a fuzzy open q-nbd of a 0.4 , but B qC and so a 0.4 / ∈ clC.But any fuzzy pre-q-nbd of a 0.4 is of the form V and pclV = 1 X qC and so a 0.4 ∈ [C] p .

Fuzzy p * -Regular Space: Some Characterizations
In this section a new type of fuzzy regularity has been introduced and studied and shown that in this space p * -closure operator and pcl operator coincide.
Definition 4.1.An fts (X, τ ) is said to be fuzzy p * -regular if for each fuzzy point x t and each fuzzy pre-q-nbd U of x t , there exists a fuzzy preopen set V in X such that x t qV ≤ pclV ≤ U .Theorem 4.1.For an fts (X, τ ), the following conditions are equivalent: (a) X is fuzzy p * -regular space.
(b) For any fuzzy set A in X, [A] p = pclA, (c) For each fuzzy point x t and each fuzzy preclosed set F with x t ∈ F , there exists a fuzzy preopen set U such that x t ∈ pclU and F ≤ U .(d) For each fuzzy point x t and each fuzzy preclosed set F such that x t ∈ F , there exist fuzzy preopen sets U and V in X such that x t qU, F ≤ V and U qV .(e) For any fuzzy set A and any fuzzy preclosed set F with A ≤ F , there exist fuzzy preopen sets U and V such that AqU, F ≤ V and U qV .(f) For any fuzzy set A and any fuzzy preopen set U with AqU , there exists a fuzzy preopen set V such that AqV ≤ pclV ≤ U .
Proof.(a) ⇒ (b): By Note 2.1, it suffices to show that [A] p ≤ pclA, for any fuzzy set A in X.
(a) the fuzzy sets 0 X and 1 X are fuzzy p * -closed sets in X, (b) for two fuzzy sets A and B in X, if A ≤ B, then [A] p ≤ [B] p , (c) the intersection of any two fuzzy p * -closed sets in X is fuzzy p * -closed in X. Proof.(a) and (b) are obvious.(c) Let A and B be any two fuzzy p * -closed sets in X.Then A = [A] p and B = [B] p .Now A B ≤ A, A B ≤ B. Then by (b), [A B] p ≤ [A] p and [ Let C and D be two fuzzy sets given by C(a) = 0.5, C(b) = 0.6, D(a) = 0.2, D(b) = 0.7.Then (C D)(a) = 0.5, (C D)(b) = 0.7.Now a 0.6 / ∈ [C] p as a 0.6 qU where U (a) = 0.41, U (b) = 0.31, but pclU = U qC. Again a 0.6 / ∈ [D] p as a 0.6 qV where V (a) = 0.7, V (b) = 0.2, but pclV = V qD.
θ and the complement of a fuzzy θ-closed set is called fuzzy θ-open.
α and the complement of fuzzy α * -closed set is called fuzzy α * -open.Result 3.1.[A] p ≤ [A] θ , for any fuzzy set A in an fts X.

Remark 3 . 1 .Remark 3 . 2 .
It is clear from the following example that [A] p = [A] θ , for any fuzzy set A in an fts X, in general.Example 3.1.Consider Example 2.1.Consider the fuzzy point a 0.51 and a fuzzy set C given by C(a) = C(b) = 0.1.Then U (a) = 0.5, U (b) = 0 being a fuzzy pre-q-nbd of a 0.51 , pclU = U qC and so a 0.51 / ∈ [C] p .But other than 1 X , B is the only fuzzy open q-nbd of a 0.51 and clB = 1 X qC.Therefore, a 0.51 ∈ [C] θ .Result 3.2.[A] p ≤ [A] α , for any fuzzy set A in an fts X. Proof.Let x t ∈ [A] p .Let U be a fuzzy α-open q-nbd of x t .Then U is a fuzzy preopen set and hence pclU qA ⇒ αclU qA ⇒ x t ∈ [A] α .It is clear from the following example that [A] p = [A] α , for any fuzzy set A in an fts X, in general.Example 3.2.Let X = {a, b}, τ = {0 X , 1 X , A, B} where A(a) = 0.5, A(b) = 0.4, B(a) = 0.7, B(b) = 0.5.Then (X, τ ) is an fts.The collection of all fuzzy α-open sets is {0 X , 1 X , A, B, V } where V ≥ B and that of fuzzy preopen sets
set.The smallest fuzzy α-closed set containing a fuzzy set A is called fuzzy α-closure p * -Closure Operator and p * -Regularity in Fuzzy Setting of A and is denoted by αclA [2], i.e., αclA =U : A ≤ U and U is fuzzy α-closedA fuzzy set A in X is fuzzy α-closed if A = αclA [2].Afuzzy set B is called a quasi neighbourhood (q-nbd, for short) of a fuzzy set A in an fts X if there is a fuzzy open set U in X such that AqU ≤ B. If, in addition, B is fuzzy open (resp., α-open) then B is called a fuzzy open (resp., α-open) q-nbd of A. In particular, a fuzzy set B in X is a fuzzy open (resp., α-open) q-nbd of a fuzzy point x t in X if x t qU ≤ B, for some fuzzy open (resp., α-open) set U in X. 2. Fuzzy p * -Closure Operator: Some Properties In this section fuzzy p * -closure operator has been introduced and studied.Let us recall a definition from [4] for ready reference.Definition 2.1 ([4]).A fuzzy set A in an fts (X, τ ) is called fuzzy preopen if A ≤ intclA.The complement of a fuzzy preopen set is called a fuzzy preclosed set.