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Separation axioms in smooth fuzzy topological spaces.

[sections]1. Introduction and Preliminaries

The concept of fuzzy set was introduced by Zadeh [10] in his classical paper. Fuzzy sets have applications in many fields such as information [6] and control [8]. In 1985, Sostak [7] introduced a new form of topological structure. In 1992, Ramadan [4] studied the concept of smooth fuzzy topological spaces. The concept of [??]-open set was discussed by Rajesh and Erdal Ekici [3]. The concept of fuzzy normal spaces was introduced by Bruce Hutton [1]. Kubiak [9] established many interesting properties of fuzzy normal spaces. The purpose of this paper is to introduce fuzzy [??]-Ti (i = 0,1, 2,1/2) spaces, [??]-normality and g-regularity in smooth fuzzy topological spaces. Also many interesting characterizations are established.

Throughout this paper, let X be a nonempty set, I = [0,1] and I0 = (0,1]. For {e I, T(x) = { for all x [member of] X.

Definition 1.1. [2] [lambda] fuzzy point xt in X is a fuzzy set taking value t [member of] [I.sub.0] at x and zero elsewhere, [x.sub.t] [member of] [lambda] iff t [less than or equal to] [lambda](x). [lambda] fuzzy set [lambda] is quasi-coincident with a fuzzy set [mu], denoted by [lambda] q [mu], if there exists x [member of] X such that [lambda](x) + [mu](x) > 1. Otherwise [lambda] [bar.q] [mu].

Definition 1.2. [7] [lambda] function T : [I.sup.X] [right arrow] I is called a smooth fuzzy topology on X if it satisfies the following conditions :

(1) T ([bar.0]) = T ([bar.1]) = 1;

(2) T([mu].sub.1] [conjuction] [[mu].sub.2]) [greater than or equal to] T([[mu].sub.1]) [conjuction] T([[mu].sub.2]) for any [[mu].sub.1], [[mu].sub.2] [member of] [I.sup.X].

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The pair (X, T) is called a smooth fuzzy topological space.

Remark 1.1. Let (X, T) be a smooth fuzzy topological space. Then, for each r [member of] I0,Tr = {[mu] [member of] [I.sup.X] : T([mu]) [greater than or equal to] r} is Chang's fuzzy topology on X.

Definition 1.3. [5] Let (X, T) be a smooth fuzzy topological space. For each [lambda] [member of] [I.sup.X], r [member of] [I.sub.0], an operator [C.sub.T] : [I.sup.X] x [I.sub.0] [right arrow] [I.sup.X] is defined as follows: [C.sub.T] ([lambda], r) = [conjuction] {[mu] : [mu] [greater than or equal to] [lambda], T([bar.1] - [mu]) [greater than or equal to] r}.

For [lambda], [mu] [member of] [I.sup.X] and r, s [member of] [I.sub.0], it satisfies the following conditions:

(1) [C.sub.T] ([bar.0], r) = [bar.0].

(2) [lambda] [less than or equal to] [C.sub.T] ([lambda],r).

(3) [C.sub.T] ([lambda],r) [disjunction] [C.sub.T] ([rho],r) = [C.sub.T] ([lambda] [disjunction] [mu],r).

(4) [C.sub.T] ([lambda],r) [less than or equal to] [C.sub.T] ([lambda],s), if r [less than or equal to] s.

(5) [C.sub.T] ([C.sub.T] ([lambda],r),r) = [C.sub.T] ([lambda],r).

Proposition 1.1. [5] Let (X, T) be a smooth fuzzy topological space. For each [lambda] [member] [I.sup.X], r [member of] [I.sub.0], an operator [I.sub.T] : [I.sup.X] x [I.sub.0] [right arrow] [I.sup.X] is defined as follows: [I.sub.T] ([lambda], r) = [disjunction] {[mu] : [mu] [less than or equal to] [lambda], T([mu]) [greater than or equal to] r}.

For [lambda], [mu] [member of] [I.sup.X] and r, s [member of] I0, it satisfies the following conditions:

(1) [I.sub.T] ([bar.1] - [lambda], r) = [bar.1] - [C.sub.T] ([lambda],r).

(2) [I.sub.T] ([bar.1],r) = [bar.1].

(3) [lambda] [greater than or equal to] [I.sub.T] ([lambda],r).

(4) [I.sub.T] ([lambda],r) [lambda] [I.sub.T] ([rho],r) = [I.sub.T] ([lambda] [conjuction] [mu],r).

(5) [I.sub.T] ([lambda],r) [greater than or equal to] [I.sub.T] ([lambda],s), ifr [less than or equal to] s.

(6) [I.sub.T] ([I.sub.T] ([lambda],r),r) = [I.sub.T] ([lambda],r).

Definition 1.4. [4] Let (X, T) be a smooth fuzzy topological space. For [lambda] [member of] [I.sup.X] and r [member of] [I.sub.0],

(1) S[C.sub.T]([lambda], r) = [conjuction] {[mu] [member of] [I.sup.X] :[mu] [greater thna or equal to] [lambda], [mu] is r--fuzzy semiclosed} is called r-fuzzy semiclosure of [lambda].

(2) [lambda] is called r-fuzzy semiclosed (briefly, r-fsc) if [lambda] [greater than or equal to] [I.sub.T] ([C.sub.T]([lambda],r),r).

(3) [lambda] is called r-fuzzy semiopen (briefly, r-fso) if [lambda] [less than or equal to] [C.sub.T] ([I.sub.T]([lambda],r),r).

Definition 1.5. [4] Let (X, T) and (Y, S) be any two smooth fuzzy topological spaces. Let f : (X, T) [right arrow] (Y, S) be a function. Then

(1) f is called fuzzy continuous iff S([mu]) [less than or equal to] T([f.sup.-1] ([mu])) for each [mu] [member of] [I.sup.Y].

(2) f is called fuzzy open iff T([lambda]) [less than or equal to] S(f ([lambda])) for each [lambda] [member of] [I.sup.X].

[sections]2. Fuzzy [??]-[T.sub.i] spaces

In this section, the concept of fuzzy [??]-[T.sub.i] (i = 0,1, 2,1/2) spaces is introduced. Interesting properties and characterizations of such spaces are discussed.

Definition 2.1. Let (X, T) be a smooth fuzzy topological space. For [lambda] [member of] [I.sup.X] and r [member of] [I.sub.0], [lambda] is called

(1) r-fuzzy [??]-closed if [C.sub.T] ([lambda], r) [less than or equal to] [mu] whenever [lambda] [less than or equal to] [mu] and [mu] is r-fuzzy semiopen. The complement of a r-fuzzy [??]-closed set is said to be a r-fuzzy [??]-open set.

(2) r-fuzzy *[??]-closed if [C.sub.T]([lambda], r) [less than or equal to] [mu] whenever [lambda] [less than or equal to] [mu] and [mu] is r-fuzzy [??]-open. The complement of a r-fuzzy *[??]-closed set is said to be a r-fuzzy *[??]-open set.

(3) r-fuzzy #g-semiclosed (briefly r-#fgs-closed) if S[C.sub.T]([lambda],r) [less than or equal to] [mu] whenever [lambda] [less than or equal to] [mu] and [mu] is r-fuzzy *[??]-open. The complement of a r-fuzzy #g-semiclosed set is said to be a r-fuzzy #gsemiopen set (briefly r-#fgs-open set ).

(4) r-fuzzy [??]-closed if [C.sub.T]([lambda], r) [less than or equal to] [mu] whenever [lambda] [less than or equal to] [mu] and [mu] is r-#fgs-open. The complement of a r-fuzzy [??]-closed set is said to be a r-fuzzy [??]-open set.

Definition 2.2. Let (X, T) be a smooth fuzzy topological space. For [lambda] [member of] [I.sub.X] and r [member of] [I.sub.0],

(1) [??]-It([lambda],r) = [disjunction] {[mu] [member of] [I.sup.X] : [mu] [less than or equal to] [lambda], [mu] is a r-fuzzy [??]-openset} is called r-fuzzy [??]-interior of [lambda].

(2) [??]-[C.sub.T]([lambda], r) = [disjunction]{[mu] [member of] [I.sup.X] :[mu] [greater thna or equal to] [lambda], [mu] is a r--fuzzy [??]--closedset} is called r-fuzzy [??]-closure of [lambda].

Definition 2.3. Let (X, T) be a smooth fuzzy topological space. For [lambda] [member of] [I.sup.X] and r [member of] [I.sub.0], [lambda] is called r-generalized fuzzy [??]-closed (briefly, r-gf[??]-closed) iff [??]-[C.sub.T]([lambda],r) [less than or equal to] [mu] whenever [lambda] [less than or equal to] [mu], [mu] [member of] [I.sup.X] is r-fuzzy [??]-open. The complement of a r- generalized fuzzy [??]-closed set is a r-generalized fuzzy [??]-open set (briefly, r-gf[??]-open).

Definition 2.4. Let (X, T) and (Y, S) be any two smooth fuzzy topological spaces. Let f : (X, T) [right arrow] (Y, S) be a function.

(1) f is called [??]-open ( resp. [??]-closed ) if for each r-fuzzy [??]-open set [lambda] [member of] [I.sup.X],f ([lambda]) [member of] [I.sup.Y] is r-fuzzy [??]-open ( resp. r-fuzzy [??]-closed ).

(2) f is called g-continuous if for each [lambda] [member of] [I.sup.Y] with S([lambda]) [greater than or equal to] r, [f.sup.-1] ([lambda]) [member of] [I.sup.X] is r-fuzzy [??]-open.

(3) f is called fuzzy [??]-irresolute if for each r-fuzzy [??]-open set [lambda] [member of] [I.sup.Y], [f.sup.- 1]([lambda]) [member of] [I.sup.X] is r-fuzzy [??]-open.

(4) f is called fuzzy [??]-homeomorphism if f is one to one, onto, fuzzy [??]-irresolute and fuzzy [??]-open.

(5) f is called gf[??]-irresolute if for each r-gf[??] closed set [lambda] [member of] [I.sup.Y], [f.sup.-1]([lambda]) [member of] [I.sup.X] is r-gf[??]-closed.

(6) f is called gf[??]-closed iff for any r-gf[??]-closed set [lambda] [member of] [I.sup.X] , [f.sup.-1] ([lambda]) is r- gf[??]-closed.

Definition 2.5. [lambda] smooth fuzzy topological space (X, T) is called

(1) Fuzzy [??]--[T.sub.o] iff for [lambda], [mu] [member of] [I.sup.X] with [lambda] [bar.q] [mu], there exists r-fuzzy [??]-open set [delta] [member of] [I.sup.X] such that either [lambda] [less than or equal to] [delta] or [mu] [less than or equal to] [delta], [lambda][bar.q][delta].

(2) Fuzzy [??]--[T.sub.1] iff for [lambda], [mu] [member of] [I.sup.X] with [lambda] [bar.q] [mu], there exist r-fuzzy [??]-open sets [delta], [eta] [member of] [I.sup.X] such that either [lambda] [less than or equal to] [delta], [mu][bar.q] [delta] or [mu] [less than or equal to] [eta], [lambda] [bar.q] [eta].

(3) Fuzzy [??]--[T.sub.2] iff for [lambda], [mu] [member of] [I.sup.X] with [lambda] [bar.q] [mu], there exist r-fuzzy [??]-open sets [delta], [eta] [member of] [I.sup.X] with [lambda] [less than or equal to] [delta], [mu] [less than or equal to] [eta] and [delta] [bar.q] [eta].

(4) Fuzzy [??]--[R.sub.0] iff [lambda][bar.q][??]--[C.sub.T]([rho],r) implies that [mu][bar.q] [??]--[C.sub.T]([lambda],r) for [lambda], [mu] [member of] [I.sup.X].

Definition 2.6. [lambda] smooth fuzzy topological space (X, T) is called fuzzy [??]--[T.sub.1/2] if every r - gf[??]-closed set is r-fuzzy [??]-closed.

Proposition 2.1. Let (X, T) be a smooth fuzzy topological space. For r [member of] [I.sub.0], the following properties hold:

(i) For all r-fuzzy [??]--open set [lambda] [member of] [I.sup.X], [lambda] q [mu] iff [lambda] q ( [??]--[C.sub.T] ([rho],r)),[mu] [member of] [I.sup.X].

(ii) [delta] q([??]--[C.sub.T]([lambda], r)) iff [lambda] q [mu] for all r-fuzzy [??]--open set [mu] [member of] [I.sup.X] with [delta] [less than or equal to] [mu], for [lambda], [delta] [member of] [I.sup.X].

Proof. (i) Let [lambda] be a r-fuzzy [??]-open set such that [lambda] q p. Since [mu] [less than or equal to] [??]--[C.sub.T]([rho],r), [lambda] q [??] [C.sub.T] ([rho],r). Conversely let [lambda] be a r-fuzzy [??]-open set such that [lambda] [bar.q] [mu]. Then [mu] [less than or equal to] [bar.1] -- [lambda], this implies that [??]--[C.sub.T](p, r) [less than or equal to] [??]--[C.sub.T](1 -- [lambda],r) = [bar.1]--[lambda]. Now,[??]--[C.sub.T](p, r) [less than or equal to] 1-[lambda]. Thus [lambda] [bar.q] [??]--[C.sub.T]([rho],r) which is a contradiction. Hence the result.

(ii) Let [delta] q ([??]--[C.sub.T]([lambda], r)). Since [delta] [less than or equal to] [mu], [mu] q ([??]--[C.sub.T]([lambda], r)). By ( i ), [mu] q [lambda] for all r-fuzzy gopen set [mu] with [delta] [less than or equal to] [mu]. Conversely suppose that [delta] q [??]--[C.sub.T]([lambda], r). Then [delta] [less than or equal to] 1 -- ([??]--[C.sub.T]([lambda], r)). Let [mu] =1 -- ([??]--[C.sub.T]([rho],r)) . Then [mu] is a r-fuzzy [??]-open set. Since [lambda] [less than or equal to] [??]--[C.sub.T]([lambda],r),p = [bar.1]--([??]--[C.sub.T] ([lambda], r)) [less than or equal to] [bar.1]--[lambda], this implies that [lambda] [bar.q] [mu], a contradiction. Hence the result.

Proposition 2.2. Let ( X, T) be a smooth fuzzy topological space. For [delta], [rho] [member of] [I.sup.X], the following statements are equivalent:

(i) (X, T) is fuzzy [??]--[R.sub.0].

(ii) If [delta] [bar.q] [lambda] = [??]--[C.sub.T]([lambda],r), [lambda] [member of] [I.sup.X] and r [member of] [I.sub.0], there exists a r-fuzzy [??]-open set [mu] such that [delta] [bar.q] [mu] and [lambda] [less than or equal to] [mu].

(iii) If [delta][bar.q] [lambda] = [??]--[C.sub.T]([lambda],r), then [??]--[C.sub.T]([delta],r) [bar.q] [lambda] = [??]--[C.sub.T]([lambda],r),[lambda] [member of] [I.sup.X] and r [member of] [I.sub.0].

(iv) If [delta][bar.q] [??]--[C.sub.T] ([rho],r) then [??]--[C.sub.T] ([delta],r) q [??]--[C.sub.T] ([rho],r),r [member of] [I.sub.0].

Proof. (i) [??] (ii) Let [delta][bar.q] [lambda] = [??]--[C.sub.T] ([lambda],r). Since [??]--[C.sub.T] ([rho],r) [less than or equal to] [??]--[C.sub.T] ([lambda],r) for each [rho] [less than or equal to] [lambda], we have [delta][bar.q] ([??]--[C.sub.T] (p, r)). By ( i ), pq ([??]--[C.sub.T] ([delta], r)). By ( ii ) of Proposition 2.1, for each [rho] [bar.q] ([??]--[C.sub.T] ([delta], r)), there exists a r-fuzzy [??]-open set [eta] such that [delta] [bar.q] [eta], [rho] [less than or equal to] [eta]. Let [mu] = [conjunction] {[eta]: [delta] [bar.q] [eta]}. Then [delta][bar.q] [mu], [lambda] [less than or equal to] [mu] for all r-fuzzy [??]-open set [mu].

(ii) [??] (iii) Let [delta] [bar.q] [lambda] = [??]--[C.sub.T]([lambda], r). By ( ii ), there exists a r-fuzzy [??]-open set [mu] [member of] [I.sup.X] such that [delta] [bar.q] [mu] and [lambda] [less than or equal to] [mu]. Since [delta] [bar.q] [mu], it follows that [delta] [less than or equal to] [bar.1]--[mu]. This implies that [??]--[C.sub.T]([delta],r) [less than or equal to] [??]--[C.sub.T](1--[mu], r) = [bar.1]--[mu] [less than or equal to] [bar.1]--[lambda]. Hence [??]--[C.sub.T]([delta], r) q [lambda] = [??]--[C.sub.T]([lambda], r).

(iii) [??](iv) Let [delta][bar.q] [??]--[C.sub.T] ([rho],r). Since [??]--[C.sub.T] ([??]--[C.sub.T] ([rho],r),r) = [??]--[C.sub.T] ([rho],r) and by ( iii ), [??]--[C.sub.T]([delta],r) [bar.q] [??]--[C.sub.T]([rho],r).

(iv) [??] (i) Let [delta][bar.q] [??]--[C.sub.T] ([rho],r). By (iv ), [??]--[C.sub.T] ([delta],r) [bar.q] [??]--[C.sub.T] ([rho],r). Since [rho] [less than or equal to] [??]--[C.sub.T] ([rho],r), [rho][bar.q] [??]--[C.sub.T]([delta],r). Hence ( X, T) is fuzzy [??]--[R.sub.0].

Proposition 2.3. Let ( X, T ) and ( Y, S) be any two smooth fuzzy topological spaces. Let f : (X,T) -- (Y,S) be a fuzzy [??]-irresolute, gf[??]-irresolute and fuzzy [??]-closed function. Then the following conditions hold:

(i) If f is injective and (Y, [delta] ) is a fuzzy [??]--[T.sub.1/2] space, then (X, T ) is a fuzzy [??]--[T.sub.1/2] space.

(ii) If f is surjective and (X, T ) is a fuzzy [??]--[T.sub.1/2] space, then (Y, S) is a fuzzy [??]--[T.sub.1/2] space.

Proof. (i) Let [lambda] [member of] [I.sup.X] be a r-gf[??]-closed set. Since f is gf[??]-closed, f ([lambda]) [member of] [I.sup.Y] is r-gf[??]-closed. Since (Y, S) is fuzzy [??]--[T.sub.1/2], f([lambda]) is r-fuzzy[??]-closed. Now, [lambda] = [f.sup.-1](f ([lambda])) is r-fuzzy [??]-closed. Hence (X, T ) is a fuzzy [??]--[T.sub.1/2] space.

(ii) Let [mu] [member of] [I.sup.Y] be a r-gf[??]-closed set. Since f is gf[??]-irresolute, [f.sup.-1]([mu]) [member of] [I.sup.X] is a r-gf[??]-closed set. Since (X, T) is a fuzzy [??]--[T.sub.1/2] space, [f.sup.-1]([mu]) is a r-fuzzy [??]-closed set. Therefore [mu] = f ([f.sup.-1]([mu])) is r-fuzzy [??]-closed. Hence (Y, S) is a fuzzy [??]-- [T.sub.1/2] space.

Proposition 2.4. Let ( X, T ) and (Y, [delta] ) be any two smooth fuzzy topological spaces. Let f : (X, T) [right arrow] (Y, S) be a fuzzy [??]-irresolute, and injective function. If (Y, S) is fuzzy [??]-- [T.sub.2] ( resp. fuzzy [??]--[T.sub.1] ), then (X, T ) is fuzzy [??]--[T.sub.2] ( resp. fuzzy [??]--[T.sub.1] ).

Proof. Let ( Y, S) be a fuzzy [??]--[T.sub.2] space. Let [[lambda].sub.1], [[lambda].sub.2] [member of] [I.sup.X] be such that [[lambda].sub.1] [bar.q] [[lambda].sub.2], then exist r-fuzzy [??]-open sets [lambda], [mu] [member of] [I.sup.Y] such that f ([[lambda].sub.1]) [less than or equal to] [lambda] and f ([[lambda].sub.2]) [less than or equal to] [mu] such that [lambda] [bar.q] [mu]. Then A [less than or equal to] [bar.1]--[mu] which implies that [f.sup.-1]([lambda]) [bar.q] [f.sup.-1]([mu]). Now, [[lambda].sub.1] [less than or equal to] [f.sup.-1]([lambda]) and [[lambda].sub.2] [less than or equal to] [f.sup.-1]([mu]). Since f is fuzzy [??]-irresolute, [f.sup.-1] ([lambda]) and [f.sup.-1]([mu]) are r-fuzzy [??]-open sets. Hence (X, T ) is a fuzzy g --[T.sub.2] space. Similarly we prove the case of fuzzy [??]--[T.sub.1] space.

[sections]3. Fuzzy [??]-normal spaces and its characterizations

In this section, the concept of fuzzy [??]-normal space is introduced. Interesting properties and characterizations of such space are discussed.

Definition 3.1. [lambda] smooth fuzzy topological space (X, T ) is said to be fuzzy [??]-normal if for every r-fuzzy [??]-closed set [lambda] and r-fuzzy [??]-open set [mu] with [lambda] [less than or equal to] [mu] there exists a [gamma] [member of] [I.sup.X] such that [lambda] [less than or equal to] [??]--[I.sub.T] ([gamma], r) [less than or equal to] [??]--[C.sub.T] ([gamma], r) [less than or equal to] [mu],r [member of] [I.sub.0].

Proposition 3.1. For any smooth fuzzy topological space (X, T) and [lambda], [mu], [delta] [member of] [I.sup.X], r [member of] [I.sub.0], the following statements are equivalent:

(i) (X, T) is fuzzy [??]-normal.

(ii) For each r-fuzzy [??]-closed set [lambda] and each r-fuzzy [??]-open set [mu] with [lambda] [less than or equal to] [mu], there exists a r-fuzzy [??]-open set [delta] such that [??]--[C.sub.T]([lambda], r) [less than or equal to] [delta] [less than or equal to] [??]--[C.sub.T]([delta], r) [less than or equal to] [mu].

(iii) For each r-gf[??]-closed set [lambda] and r-fuzzy [??]-open set [mu] with [lambda] [less than or equal to] [mu], there exists a r-fuzzy [??]-open set [delta] such that [??]--[C.sub.T]([lambda],r) [less than or equal to] [delta] [less than or equal to] [??]--[C.sub.T]([delta],r) [less than or equal to] [mu].

Proof. (i) [??] (ii) The proof is trivial.

(ii) [??] (iii) Let [lambda] be any r-gf[??]-closed set and [mu] be any r-fuzzy [??]-open set such that [lambda] [less than or equal to] [mu]. Since [lambda] is r-gf[??]-closed, [??]--[C.sub.T]([lambda], r) [less than or equal to] [mu]. Now, [??]--[C.sub.T]([lambda], r) is r-fuzzy [??]-closed and [mu] is r-fuzzy gopen. By (ii), there exists a r-fuzzy [??]-open set [delta] such that [??]--[C.sub.T]([lambda], r) [less than or equal to] [delta] [less than or equal to] [??]--[C.sub.T]([delta], r) [less than or equal to] [mu].

(iii) [??] (i) The proof is trivial.

Proposition 3.2. Let (X, T ) and (Y, S) be any two smooth fuzzy topological spaces. If f : (X, T) [right arrow] (Y, S) is fuzzy [??]-homeomorphism and (Y, S) is fuzzy [??]-normal, then (X, T) is fuzzy [??]-normal.

Proof. Let [lambda] [member of] [I.sup.X] be any r-fuzzy [??]-closed set and [mu] [member of] [I.sup.X] be any r-fuzzy [??]-open set such that [lambda] [less than or equal to] [mu] where r [member of] [I.sub.0]. Since f is fuzzy [??]- homeomorphism, it is also fuzzy [??]-closed. Hence f ([lambda]) [member of] [I.sup.Y] is r-fuzzy [??]-closed. Since f is fuzzy [??]-open, f ([mu]) [member of] [I.sup.Y] is r-fuzzy [??]-open. Since (Y, S) is fuzzy [??]-normal, there exists a 7 [member of] [I.sup.Y] such that f ([lambda]) [less than or equal to] [??]--[I.sub.T]([gamma], r) [less than or equal to] [??]--[C.sub.T]([gamma], r) [less than or equal to] f ([rho]). Now, [f.sup.-1](f([lambda])) = [lambda] [less than or equal to][f.sup.-1]([??]--[I.sub.T]([gamma],r)) [less than or equal to] f-1([??]--[C.sub.T]([gamma],r)) [less than or equal to] f-1(f([mu])) = [mu]. That is, A [less than or equal to] [??]--[I.sub.T](f-1([gamma]),r) [less than or equal to] [??]--[C.sub.T](f-1([gamma]),r) [less than or equal to] [mu]. Therefore (X, T) is fuzzy [??]-normal.

Proposition 3.3. Let (X, T ) and (Y, S) be any two smooth fuzzy topological spaces. If f : (X, T) -- (Y, S) is fuzzy [??]-homeomorphism and (X, T) is a fuzzy [??]-normal space, then (Y, S) is fuzzy [??]-normal.

Proof. Let [lambda] [member of] [I.sup.Y] be any r-fuzzy [??]-closed set and [mu] [member of] [I.sup.Y] be any r-fuzzy [??]-open set such that [lambda] [less than or equal to] [mu] where r [member of] [I.sub.0]. Since f is fuzzy [??]-irresolute, f-1([lambda]) is r-fuzzy [??]-closed and [f.sup.-1]([mu]) [member of] [I.sup.Y] is r-fuzzy [??]-open. Since (X, T) is fuzzy [??]-normal, there exists a 7 [member of] [I.sup.X] such that f-1([lambda]) [less than or equal to] [??]--[I.sub.T]([gamma],r) [less than or equal to] [??]--[C.sub.T]([gamma],r) [less than or equal to] [f.sup.-1]([mu]). Now, f(f-1([lambda])) = [lambda] [less than or equal to] f([??]--[I.sub.T]([gamma],r)) [less than or equal to] f([??]--[C.sub.T]([gamma],r)) [less than or equal to] f([f.sup.-1]([mu])) = p. That is, [lambda] [less than or equal to] [??]--[I.sub.T](f([gamma]),r) [less than or equal to] [??]--[C.sub.T](f([gamma]),r) [less than or equal to] [mu]. Therefore (Y, S) is fuzzy [??]-normal.

Proposition 3.4. Let (X, T) be a smooth fuzzy topological space which is also a fuzzy [??]-normal space. Let [{[[lambda].sub.i]}.sub.i[member of]J [I.sup.X] and [{[[mu].sub.j]}.sub.j[member of]J] [member of] [I.sup.X] .If there exist [lambda], [mu] [member of] [I.sup.X] such that [??]--[C.sub.T]([A.sub.i],r) [less than or equal to] [??]--[C.sub.T]([lambda],r) [less than or equal to] [??]--[I.sub.T]([[mu].sub.j],r) and [??]--[C.sub.T]([lambda].sub.i],r) [less than or equal to] [??]--[I.sub.T]([mu],r) [less than or equal to] [??]--[I.sub.T](pj,r) for all i,j = 1, 2, * * *, and r [member of] [I.sub.0], then there exists [gamma] [member of] [I.sup.X] such that

[??]--[C.sub.T] ([lambda].sub.i],r) [less than or equal to] [??]--[I.sub.T] ([gamma],r) [less than or equal to] [??]--[C.sub.T] ([gamma],r) [less than or equal to] [??]--[I.sub.T] ([[mu].sub.j] ,r), for all i,j = 1, 2, *** .

Proof. First, we shall show by induction that for all n [greater than or equal to] 2 there exists a collection {[[gamma].sub.i], [[delta].sub.i]/1 [less than or equal to] i [less than or equal to] n} contained in [I.sup.X] such that the conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

hold for all i,j = 1, 2, . ...n-1. Clearly (S2) follows at once from the fuzzy [??]-normality of (X, T). Now, suppose that for n [greater than or equal to] 2, Yi, Si G [I.sup.X](i < n) such that (Sn) holds. Since [??]-[C.sub.T](An,r) [less than or equal to] [??]-[C.sub.T]([lambda],r) [less than or equal to] 9 -It(Sj,r)(j < n) and [??]-[C.sub.T](K,r) [less than or equal to] 9 -It([rho],r) by fuzzy [??]-normality of (X, T), there exists Yn G [I.sup.X] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, since [??]-[C.sub.T]([lambda],r) [less than or equal to] [??]-[I.sub.T]([[mu].sub.n],r) and [??]-[C.sub.T]([[gamma].sub.i],r) [less than or equal to] [??]-[I.sub.T]([[mu].sub.n],r)(i [less than or equal to] n), there exists [[dlta[.sub.n] [member of] [I.sup.X] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus ([S.sub.n+1]) holds.

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [??]-[C.sub.T]([lambda].sub.i],r) [less than or equal to] [??]-[I.sub.T]([[gamma].sub.i], r) [less than or equal to] [??]-[I.sub.T] ([gamma], r) for all i = 1, 2, ***. Since

[??]-[C.sub.T] ([[gamma].sub.i],r) [less than or equal to] [??]-[I.sub.T] ([[delta].sub.j] ,r)(i,j = 1, 2, *** ), [[gamma].sub.i] [less than or equal to] [[delta].sub.j], so that [??]-[C.sub.T] ([gamma],r) [less than or equal to] [??]-[C.sub.T] ([[delta].sub.j], r) [less than or equal to] [??]-[I.sub.T]([[mu].sub.j], r) for all j = 1, 2, ***. This proves the result.

Proposition 3.5. Let (X, T) be a smooth fuzzy topological space which is also a fuzzy [??]-normal space. If [{[[lambda].sub.q]}.sub.q[member of]Q] and [{[[mu].sub.q]}.sub.q[member of]Q] are monotone increasing collections of respectively, fuzzy [??]-closed and fuzzy "-open subsets of (X, T) (Q is the set of all rational numbers ) such that [[lambda].sub.q] [less than or equal to] [[mu].sub.s] whenever q [less than or equal to] s, then there exists a collection [{[[gamma].sub.q]}.sub.q[member of]Q] [member of] [I.sup.X] such that

[[lambda].sub.q] [less than or equal to] [??]-[I.sub.T]([[gamma].sub.s],r), [??]-[C.sub.T]([Y.sub.q],r) [less than or equal to] [??]-[I.sub.T]([Y.sub.s],r) and [??]-[C.sub.T]([Y.sub.q],r) [less than or equal to] [[mu].sub.s]

whenever q < s.

Proof. Let us arrange into a sequence qn of all rational numbers (without repetitions). For every n [greater than or equal to] 2 we shall define inductively a collection {[[gamma].sub.qi] /1 [less than or equal to] i [less than or equal to] n} [member of] [I.sup.X] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all 1 [less than or equal to] i,j [less than or equal to] n. It is clear that the countable collections {[[lambda].sub.q]/q < [q.sub.1]} and {[mu].sub.q]/q > [q.sub.1]} together with [[lambda].sub.q1] and [[mu].sub.q1] satisfy all hypotheses of Proposition 3.4, so that there exists S1 G [I.sup.X] such that [[lambda].sub.q] [less than or equal to] [??]-[I.sub.T] ([[delta].sub.1], r) for all q [less than or equal to] q1 and [??]-[C.sub.T]([[delta].sub.1],r) [less than or equal to] [[mu].sub.q] for all q > [q.sub.1].

Letting [[gamma.sub.q1] = [[delta].sub.1], we get ([S.sub.2]). Assume that the fuzzy subsets [[gamma.sub.q1] are already defined for i < n and satisfy ([S.sub.n]). Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then [??]-[C.sub.T] ([[gamma.sub.qi] ,r) [less than or equal to] [??]-[C.sub.T] ([lambda],r) [less than or equal to] [??]-[I.sub.T] ([Y.sub.qj] ,r), and [??]--[C.sub.T] ([[gamma.sub.qi] ,r) [less than or equal to] [??]-[I.sub.T] ([mu],r) [less than or equal to] [??]- [I.sub.T] ([[gamma].sub.qj],r)

whenever [q.sub.i] < [q.sub.n] < [q.sub.j] (i,j < n) as well as [[lambda].sub.q] [less than or equal to] [??]- [C.sub.T]([lambda],r) [less than or equal to] [[mu].sub.s] and [[lambda].sub.q] [less than or equal to] [??]-[I.sub.T]([mu],r) [less than or equal to] [[mu].sub.s] whenever q < qn < s. This shows that the countable collections {[[gamma.sub.qi]/i <, [q.sub.i] < [q.sub.n]} [disjunction] {[[lambad].sub.q]/q < [q.sub.n]} and {[Y.sub.qj]/j < n, [q.sub.j] > qn} [disjunction] {[[mu].sub.q]/q > [q.sub.n]} together with [lambda] and [mu] satisfy all hypotheses of Proposition 3.5. Hence there exists a [[delta].sub.n] [member of] [I.sup.x] such that

Aq [less than or equal to] 9 -It(Sn, r), if q [less than or equal to] qn

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 1 [less than or equal to] i,j [less than or equal to] n - 1. Letting [Y.sub.qn] = [[delta].sub.n] we obtain fuzzy subsets [[gamma.sub.q1] ,[[gamma].sub.q2], *** ,[Y.sub.qn] that satisfy the result ([S.sub.n+1]). Therefore the collection {[[gamma.sub.q1]/i = 1, 2, *** .} has the required properties. This completes the proof.

[sections]4. Fuzzy [??]-regular space and its characterizations

In this section, the concept of fuzzy [??]-regular space is introduced. Some interesting characterizations are established.

Definition 4.1. [lambda] smooth fuzzy topological space (X, T) is called a fuzzy [??]-regular space if for every r- fuzzy [??]-closed set [lambda] and each [alpha] [member of] [I.sup.X] with [alpha] [??] [lambda], there exist [mu],[delta] [member of] [I.sup.X] with T ([mu]) [greater than or equal to] r, T ([delta]) [greater than or equal to] r and [delta][bar.q] [mu] such that a [less than or equal to] [delta], [lambda] [less than or equal to] [mu].

Proposition 4.1. Let (X, T) be a smooth fuzzy topological space. Then the following statements are equivalent:

(i) (X, T) is fuzzy [??]-regular.

(ii) For each a G [I.sup.X] and r-fuzzy [??]-open set [lambda] with a q [lambda] there exists a [delta] G [I.sup.X] with T([delta]) [greater than or equal to] r, [alpha] [less than or equal to] [delta] such that [C.sub.T]([delta], r) [less than or equal to] A.

Proof. (i) [??] (ii) Let [lambda] be any r-fuzzy [??]-open set with [alpha] [??] [gamma]. By hypothesis, there exist [mu], [delta] [member of] [I.sup.X] with T([mu]) [greater than or equal to] r, T([delta]) [greater than or equal to] r and [delta] [bar.q] [mu] such that [bar.1]-[lambda] [less than or equal to] [mu] and [alpha] [less than or equal to] [delta]. Since [delta] [less than or equal to] [bar.1] - [mu], [C.sub.T] ([delta],r) [less than or equal to] [C.sub.T]([bar.1]-[mu],r) = [bar.1]--[mu]. But [bar.1]-[lambda] [less than or equal to] [mu] gives! -- [mu] [less than or equal to] A. That is, [C.sub.T]([delta],r) [less than or equal to] [lambada].

Hence the result.

(ii) [??] (i) Let y be any r-fuzzy [??]-closed set with a [??] [gamma] for any [alpha] [member of] [I.sup.X]. Now, [bar.1]- [gamma] is r-fuzzy [??]-open. By hypothesis, there exists a [delta] G [I.sup.X] with T([delta]) [greater than or equal to] r, [alpha] [less than or equal to] [delta] such that [C.sub.T]([delta], r) [less than or equal to] [bar.1]-[gamma]. Then [gamma] [less than or equal to] [bar.1]-[C.sub.T]([delta],r). Now, [delta] [less than or equal to] [bar.1]- ([bar.1]-[C.sub.T]([delta],r)) such that [alpha] [less than or equal to] [delta] and y [less than or equal to] [bar.1]-[C.sub.T]([delta],r). Therefore (X, T) is fuzzy [??]-regular.

Proposition 4.2. Let (X, T) be a smooth fuzzy topological space. Then (X, T) is fuzzy [??]-regular iff for every r- fuzzy [??]-closed set [lambda] [member of] [I.sup.X] and [alpha] [member of] [I.sup.X] with a [??] [lambda], there exist [mu], [delta] [member of] [I.sup.X] with T([mu]) [greater than or equal to] r, T([delta]) [greater than or equal to] r such that a [less than or equal to] [delta], [lambda] [less than or equal to] [mu], then [mu] q [C.sub.T]([delta],r) where r [member of] [I.sub.0].

Proof. Let (X, T ) be a fuzzy [??]-regular space. Let [lambda] be any r-fuzzy [??]-closed set and [alpha] be such that [alpha] [??] [lambda]. Since (X, T) is fuzzy [??]-regular, there exist [mu], [delta] with T([mu]) [lambda] r; T([delta]) [lambda] r; [delta] [bar.q] [mu] such that [alpha] [less than or equal to] [delta], [lambda] [less than or equal to] [mu]. Now, [delta] q [mu] implies that [C.sub.T] ([delta], r) [less than or equal to] [C.sub.T] (1 - [mu], r) = 1 - [mu]. That is, [mu] [bar.q] [C.sub.T] ([delta], r). Hence the result. Converse part is trivial.

Proposition 4.3. Let (X, T) and (Y, S ) be any two smooth fuzzy topological spaces. If f : (X; T) [right arrow] (Y; S) is bijective, fuzzy [??]-irresolute, fuzzy open and if (X, T) is a fuzzy [??]-regular space, then (Y, S) is fuzzy [??]-regular.

Proof. Let [lambda] [member of] [I.sup.Y] be any r-fuzzy [??]-closed set and [beta] [member of] [I.sup.Y] be such that [beta] [??] [lambda], r [member of] [I.sub.0]. Since f is fuzzy [??]-irresolute, [f.sup.-1]([lambda]) [member of] [I.sup.X] is r-fuzzy [??] -closed. Let f([alpha]) = [beta] for any [alpha] [member of] [I.sup.X]. Since f is bijective, [alpha] = [f.sup.-1]([beta]). Since (X, T) is fuzzy [??]-regular and [alpha] [??] [f.sup.-1]([lambda]) there exist [mu][delta] 2 [I.sup.X] with T([mu]) [greater than or equal to] r, T([delta]) [greater than or equal to] r and [delta] q [mu] such that [alpha] [less than or equal to] [delta] and [f.sup.-1]([lambda]) [less than or equal to] [mu]. Since f is fuzzy open and bijective, f([alpha]) [less than or equal to] f([delta]) implies that [beta] [less than or equal to] f([delta]); [lambda] [less than or equal to] f([mu]) and S(f([delta])) [greater than or equal to] r, S(f([mu])) [lambda] r with f([delta]) [bar.q] f([mu]). Hence (Y, S) is fuzzy [??]-regular.

Proposition 4.4. Let (X, T ) and (Y, S ) be any two smooth fuzzy topological spaces. If f : (X, T) [right arrow] (Y; S) is fuzzy [??]-closed, fuzzy continuous, injective and ( Y, S ) is fuzzy [??]-regular then (X, T ) is fuzzy [??]-regular.

Proof. Let [lambda] [member of] [I.sup.X] be any r-fuzzy [??]-closed set and [alpha] [member of] [I.sup.X] be such that [alpha] [??] [lambda], r [member of] [I.sub.0]. Since f is fuzzy [??]-closed, f([lambda]) [member of] [I.sup.Y] is r-fuzzy [??]-closed and f([alpha]) [??] f([lambda]). Since (Y, S) is fuzzy [??]-regular, there exist [mu]; [delta] [member of] [I.sup.Y] with S([mu]) [lambda] r, S([delta]) [lambda] r and [delta] [bar.q] [mu] such that f([alpha]) [less than or equal to] [mu] and f([lambda]) [less than or equal to] [delta]. Since f is fuzzy continuous, [f.sup.-1]([mu]); [f.sup.-1]([delta]) [member of] [I.sup.X] with T([f.sup.-1]([mu])) [lambda] r and T([f.sup.-1]([delta])) [lambda] r. Also, [alpha] [less than or equal to] [f.sup.-1]([mu]); [lambda] [less than or equal to] [f.sup.-1]([delta]) and [f.sup.-1]([delta]) [bar.q] [f.sup.-1]([mu]). Therefore (X, T) is fuzzy [??]-regular.

Proposition 4.5. Let (X, T) be a smooth fuzzy topological space. Then the following statements are equivalent:

(i) (X, T) is fuzzy [??]-regular.

(ii) For every r-fuzzy [??]-open set [lambda] such that [alpha] [less than or equal to] [lambda] there exists a [gamma] [member of] [I.sup.X] with T([gamma]) [lambda] r such that [alpha] [less than or equal to] [gamma] [less than or equal to] [C.sub.T] ([gamma]; r) [less than or equal to] [lambda].

(iii) For every r-fuzzy [??]-open set [lambda] such that [alpha] [less than or equal to] [lambda] there exists a [delta] [member of] [I.sup.X] with T([delta]) [greater than or equal to] r and [delta] = [I.sub.T] ([DELTA], r), T(1 - [DELTA]) [lambda] r such that [alpha] [less than or equal to] [delta] [less than or equal to] [C.sub.T] ([delta], r) [less than or equal to] [lambda].

(iv) For every r-fuzzy [??]-closed set [mu] such that [alpha] [??] [mu] there exist [gamma] and [lambda] with T([gamma]) [greater than or equal to] r and T([lambda]) [greater than or equal to] r such that [alpha] [less than or equal to] [gamma], [mu] [less than or equal to] [lambda] with [C.sub.T] ([gamma], r) q [C.sub.T] ([lambda], r).

Proof. (i) [??] (ii) Let [lambda] be a r-fuzzy [??]-open set such that [alpha] [less than or equal to] [lambda]. Then [bar.1] - [lambda] is a r-fuzzy [??]-closed set such that [alpha] [??] 1 - [lambda]. Since (X, T) is [??]-regular, there exist [gamma], [delta] [member of] [I.sup.X] with T([gamma]) [greater than or equal to] r, T([delta]) [greater than or equal to] r and [gamma] [bar.q] [delta] such that [alpha] [less than or equal to] [gamma], 1 - [lambda] [less than or equal to] [delta]. Since [gamma] [bar.q] [delta], [gamma] [less than or equal to] 1 - [delta]. Hence [C.sub.T] ([gamma]; r) [less than or equal to] [C.sub.T] (1 - [delta]; r) = 1 - [delta]. But 1 - [delta] [less than or equal to] [lambda]. Therefore [alpha] [less than or equal to] [gamma] [less than or equal to] [C.sub.T] ([gamma]; r) [less than or equal to] [lambda].

(ii) [??] (iii) Let [lambda] be a r-fuzzy [??]-open set such that [alpha] [less than or equal to] [lambda]. By ( ii ), there exists a [gamma] 2 [I.sup.X] with T([gamma]) [greater than or equal to] r such that [alpha] [less than or equal to] [gamma] [less than or equal to] [C.sub.T] ([gamma], r) [less than or equal to] [lambda]. Let [delta] = [I.sub.T] ([DELTA]; r) where [DELTA] = [C.sub.T] ([gamma], r). Now, [alpha] [less than or equal to] [gamma] [less than or equal to] [I.sub.T] ([DELTA]; r) [less than or equal to] [C.sub.T] ([DELTA]; r) [less than or equal to] [lambda]. Also, then [alpha] [less than or equal to] [delta] [less than or equal to] [C.sub.T] ([delta]; r) = [C.sub.T] ([I.sub.T] ([DELTA]; r); r) [less than or equal to] [C.sub.T] ([DELTA]; r) = [C.sub.T] ([C.sub.T] ([gamma]; r); r) = [C.sub.T] ([gamma]; r) [less than or equal to] [lambda]. Thus [alpha] [less than or equal to] [delta] [less than or equal to] [C.sub.T] ([delta]; r) [less than or equal to] [lambda].

(iii) [??] (iv) Let [mu] be a r-fuzzy [??]-closed set with [alpha] [mu]. Then [bar.1] - [mu] is a r-fuzzy [??]-open set with [alpha] [less than or equal to] [bar.1]-[mu]. By (iii), there exists a [delta] [member of] [I.sup.X] with T([delta]) [lambda] r such that [alpha] [less than or equal to] [delta] [less than or equal to] [C.sub.T] ([delta], r) [less than or equal to] [bar.1]-[mu] where [delta] = [I.sub.T] ([DELTA], r) for some [DELTA] [member of] [I.sup.X] with T(1 - [DELTA]) [greater than or equal to]] r. Again by hypothesis there exists a [gamma] [member of] [I.sup.X] such that [alpha] [less than or equal to] [gamma] [less than or equal to] [C.sub.T] ([gamma], r) [less than or equal to] [delta]. Let [lambda] = [bar.1] - [C.sub.T] ([delta], r). Then [alpha] [less than or equal to] [gamma], [mu] [less than or equal to] [lambda] with [lambda] [less than or equal to] [bar.1] - [delta]. Now, [C.sub.T] ([lambda], r) [less than or equal to] 1 - [delta] [less than or equal to] 1 - [C.sub.T] ([gamma]; r). Thus [C.sub.T] ([gamma], r) q [C.sub.T] ([lambda], r).

(iv) [??] (i) The proof is trivial.

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B. Amudhambigai, M. K. Uma and E. Roja

Department of Mathematics Sri Sarada College for Women, Salem-16 Tamil Nadu, India E-mail: rbamudha@yahoo.co.in
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