© Hindawi Publishing Corp. α-COMPACTNESS IN SMOOTH TOPOLOGICAL SPACES

We introduce the concepts of smooth α-closure and smooth α-interior of a fuzzy set which are generalizations of smooth closure and smooth interior of a fuzzy set defined by Demirci (1997) and obtain some of their structural properties.

1. Introduction.Badard [1] introduced the concept of a smooth topological space which is a generalization of Chang's fuzzy topological space [2].Many mathematical structures in smooth topological spaces were introduced and studied.In particular, Gayyar et al. [5] and Demirci [3,4] introduced the concepts of smooth closure and smooth interior of a fuzzy set and several types of compactness in smooth topological spaces and obtained some properties of them.
In this paper, we define the smooth α-closure and smooth α-interior of a fuzzy set and investigate some of their properties.In fact, the smooth αclosure and smooth α -interior of a fuzzy set coincide with the smooth closure and smooth interior of a fuzzy set defined in [3] when α = 0. We also introduce the concepts of several types of α-compactness using smooth α-closure and smooth α-interior of a fuzzy set and investigate some of their properties.

Preliminaries.
In this section, we give some notations and definitions which are to be used in the sequel.Let X be a set and let I = [0, 1] be the unit interval of the real line.Let I X denote the set of all fuzzy sets of X.Let 0 X and 1 X denote the characteristic functions of φ and X, respectively.
A smooth topological space (s.t.s.) [6] is an ordered pair (X, τ), where X is a nonempty set and τ : I X → I is a mapping satisfying the following conditions: (1) τ(0 X ) = τ(1 X ) = 1; (2) for all A, B ∈ I X , τ(A ∩ B) ≥ τ(A) ∧ τ(B); (3) for every subfamily {A i : i ∈ J} ⊆ I X , τ(∪ i∈J A i ) ≥ ∧ i∈J τ(A i ).Then the mapping τ : I X → I is called a smooth topology on X.The number τ(A) is called the degree of openness of A.
A mapping τ * : I X → I is called a smooth cotopology [6] if and only if the following three conditions are satisfied: ( (3) for every subfamily If τ is a smooth topology on X, then the mapping τ * : I X → I, defined by τ * (A) = τ(A c ) where A c denotes the complement of A, is a smooth cotopology on X.Conversely, if τ * is a smooth cotopology on X, then the mapping τ : I X → I, defined by τ(A) = τ * (A c ), is a smooth topology on X [6].
For the s.t.s.(X, τ) and α ∈ [0, 1], the family τ α = {A ∈ I X : τ(A) ≥ α} defines a Chang's fuzzy topology (CFT) on X [2].The family of all closed fuzzy sets with respect to τ α is denoted by τ * α and we have Demirci [3] introduced the concepts of smooth closure and smooth interior in smooth topological spaces as follows.
Let (X, τ) be an s.t.s. and A ∈ I X .Then the τ-smooth closure (resp., τ-smooth interior) of A, denoted by Ā (resp.,A o ), is defined by Ā Let (X, τ) and (Y , σ ) be two smooth topological spaces.A function f : X → Y is called smooth continuous with respect to τ and σ [6] if and only if A function f : X → Y is smooth continuous with respect to τ and σ if and only if τ * (f −1 (A)) ≥ σ * (A) for every A ∈ I Y .A function f : X → Y is weakly smooth continuous with respect to τ and σ if and only if σ * (A) > 0 ⇒ τ * (f −1 (A)) > 0 for every A ∈ I Y [6].
A function f : X → Y is called smooth preserving (resp., strict smooth preserving) with respect to τ and σ [5] if and only if is a smooth preserving function (resp., a strict smooth preserving function) with respect to τ and σ , then , strict smooth open preserving) with respect to τ and σ [5] if and only if 3. Smooth α-closure and smooth α-interior.In this section, we introduce the concepts of smooth α-closure and smooth α-interior of a fuzzy set in smooth topological spaces and investigate some properties of them.Definition 3.1.Let (X, τ) be an s.t.s., α ∈ [0, 1), and Proof.(a) and (b) follow directly from Definition 3.1.
The proof is similar to the proof of (c).
Proof.(a) From Definition 3.1, we have Proof.(d) For every A, B ∈ I X , we have (3.2) Proof.The proof is similar to the proof of Theorem 3.4.

Types of smooth α-compactness.
In this section, we introduce the concepts of several types of smooth α-compactness in smooth topological spaces and investigate some properties of them.Definition 4.1 [5].An s.t.s.(X, τ) is called smooth compact if and only if for every family Theorem 4.2 [4].Let (X, τ) and (Y , σ ) be two smooth topological spaces and f : X → Y a surjective weakly smooth continuous function with respect to τ and σ .If (X, τ) is smooth compact, then so is (Y , σ ).Definition 4.3.Let α ∈ [0, 1).An s.t.s.(X, τ) is called smooth nearly αcompact if and only if for every family {A i : i ∈ J} in {A ∈ I X : τ(A) > 0} covering X, there exists a finite subset J 0 of J such that Definition 4.6.A smooth topology τ : I X → I on X is called monotonic increasing (resp., monotonic decreasing) if and only if Theorem 4.7.Let (X, τ) be an s.t.s., α ∈ [0, 1), and τ a monotonic decreasing smooth topology.If (X, τ) is smooth compact, then (X, τ) is smooth nearly α-compact.
Proof.Let (X, τ) be a smooth compact s.t.s.Then for every family {A i : i ∈ J} in {A ∈ I X : τ(A) > 0} covering X, there exists a finite subset J 0 of J such that ∪ i∈J 0 A i = 1 X .Since τ(A i ) > 0 for each i ∈ J, we have A i = (A i ) o α for each i ∈ J by Theorem 3.6.Since τ is monotonic decreasing and A i ⊆ (A i ) α for each i ∈ J, we have τ(A i ) ≥ τ((A i ) α ) for each i ∈ J. Hence from Theorem 3.2, we have Proof.Let (X, τ) be a smooth nearly α-compact s.t.s.Then for every family {A i : i ∈ J} in {A ∈ I X : τ(A) > 0} covering X, there exists a finite subset Theorem 4.9.Let (X, τ) and (Y , σ ) be two smooth topological spaces, α ∈ [0, 1), and f : X → Y a surjective, weakly smooth continuous, and strict smooth α-preserving function with respect to τ and σ .If (X, τ) is smooth almost αcompact, then so is (Y , σ ).

Proof. Let {A
preserving with respect to τ and σ , from Theorem 3.13 we have We obtain the following corollary from Theorems 4.8 and 4.9.

Proof. Let {A
From the surjectivity of f we have 1 Y = f (1 X ) = f (∪ i∈J 0 ((f −1 (A i )) α ) o α ) = ∪ i∈J 0 f (((f −1 (A i )) α ) o α ).Since f : X → Y is strict smooth open α-preserving with respect to τ and σ , from Theorem 3.14 we have f (((f −1 (A i )) α ) o α ) ⊆ (f ((f −1 (A i )) α )) o α for each i ∈ J. Since f : X → Y is strict smooth α -preserving with respect to τ and σ , from Theorem 3.13 we have (f −1 (A i )) α ⊆ f −1 ((A i ) α ) for each i ∈ J. Hence, we have Proof.Let {A i : i ∈ J} be a family in {A ∈ I X : σ (A) > 0} covering X, that is, ∪ i∈J A i = 1 X .Since (X, τ) is smooth α-regular, A i = ∪ j i ∈J i {K j i ∈ I X : τ(K j i ) ≥ τ(A i ), (K j i ) α ⊆ A i } for each i ∈ J. Since ∪ i∈J A i = ∪ i∈J [∪ j i ∈J i K j i ] = 1 X and (X, τ) is smooth almost α-compact, there exists a finite subfamily {K l ∈ I X : τ(K l ) > 0, l ∈ L} such that ∪ l∈L (K l ) α = 1 X .Since for each l ∈ L there exists i ∈ J such that (K l ) α ⊆ A i , we have ∪ i∈J 0 A i = 1 X , where J 0 is a finite subset of J. Hence (X, τ) is smooth compact.