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Original Article International Journal of Fuzzy Logic and Intelligent Systems Vol. 14, No. 3, September 2014, pp. 231-239 http://dx.doi.org/10.5391/IJFIS.2014.14.3.231 ISSN(Print) 1598-2645 ISSN(Online) 2093-744X Closure, Interior and Compactness in Ordinary Smooth Topological Spaces Jeong Gon Lee, Kul Hur, and Pyung Ki Lim Division of Mathematics and Informational Statistics and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 570-749, Korea Abstract It presents the concepts of ordinary smooth interior and ordinary smooth closure of an ordinary subset and their structural properties. It also introduces the notion of ordinary smooth (open) preserving mapping and addresses some their properties. In addition, it develops the notions of ordinary smooth compactness, ordinary smooth almost compactness, and ordinary near compactness and discusses them in the general framework of ordinary smooth topological spaces. Keywords: Ordinary smooth topological space, Ordinary smooth closure (resp. interior), Ordinary smooth continuity, Ordinary smooth preserving, Ordinary smooth (resp. almost and near) compactness 1. Received: Sep. 18, 2012 Revised : Jun. 17, 2014 Accepted: Jun. 19, 2014 Correspondence to: Pyung Ki Lim ([email protected]) ©The Korean Institute of Intelligent Systems cc This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/ by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited. 231 | Introduction By considering the degree of openness of fuzzy sets, Badard [1] introduced the concept of a smooth topological space as a generalization of a classical topology as well as a Chang’s fuzzy topology [2]. Hazra et al. [3], Chattopadhyay et al. [4], Demirci [5], and Ramadan [6] have investigated the smooth topological spaces in the various aspects. El Gayyar et al. [7] showed how the concepts of closure, interior, subspace, almost and near compactness can be smoothed while obtaining more general structures with even nice properties. Ying [8, 9] presented the notion of a fuzzifying topology (called an ordinary smooth topology (OST) by Lim et al. [10]) with the consideration of the degree of openness of ordinary subsets, and discussed some of its properties. Cheong et al. [11] constructed the collection OST(X) of all ordinary smooth topologies on a set X and investigated it in the perspective of a lattice. Here we first introduce the concepts of ordinary smooth interior and ordinary smooth closure of an ordinary subset and we investigate some of their structural properties. We also present the notions of ordinary smooth (open) preserving mapping and some their properties. In addition, we develop the notions of ordinary smooth compactness, ordinary smooth almost compactness, and ordinary near compactness and examine them in the general framework of ordinary smooth topological spaces. 2. Preliminaries Let 2 = {0, 1} and let 2X denote the set of all ordinary subsets of a set X. Definition 2.1 [10]. Let X be a nonempty set. Then a mapping τ : 2X → I is called an http://dx.doi.org/10.5391/IJFIS.2014.14.3.231 ordinary smooth topology (in short, ost) on X or a gradation of openness of ordinary subsets of X if τ satisfies the following axioms: (OST1 ) τ (∅) = τ (X) = 1. (OST2 ) τ (A ∩ B) ≥ τ (A) ∧ τ (B), ∀A, B ∈ 2X . S V (OST3 ) τ ( α∈Γ Aα ) ≥ α∈Γ τ (Aα ), ∀{Aα } ⊂ 2X . The pair (X, τ ) is called an ordinary smooth topological space (in short, osts). We will denote the set of all ost’s on X as OST(X). Remark 2.2. Ying [8] called the mapping τ : 2X → I [resp. τ : I X → 2 and τ : I X → I] satisfying the axioms in Definition 2.1 as a fuzzyfying topology [resp. fuzzy topology and bifuzzy topology] on X. Remark 2.3. If I = 2, then Definition 2.1 coincides with the known definition of classical topology. Definition 2.7 [10]. Let (X, τ ) be an osts and let r ∈ I. Then we define two ordinary subsets of X as follows : [τ ]r = {A ∈ 2X : τ (A) ≥ r} and [τ ]∗r = {A ∈ 2X : τ (A) > r}. We call these the r−level set and the strong r-level set of τ , respectively. It is clear that [τ ]0 = 2X , the classical discrete topology on X and [τ ]∗1 = ∅. Also it can be easily seen that [τ ]∗r ⊂ [τ ]r for each r ∈ I. Result 2.B [10, Proposition 2.10]. Let (X, τ ) be an osts and let T (X) be the set of all classical topologies on X. Then : (a) [τ ]r ∈ T(X), ∀r ∈ I. (a)0 [τ ]∗r ∈ T(X), ∀r ∈ I1 . (b) For any r, s ∈ I, if r ≤ s, then [τ ]s ⊂ [τ ]r and [τ ]∗s ⊂ [τ ]∗r . \ (c) [τ ]r = [τ ]s , ∀r ∈ I0 . s<r Definition 2.4 [10]. Let X be a nonempty set. Then a mapping C : 2X → I is called an ordinary smooth cotopology (in short, osct) on X or a gradation of closedness of ordinary subsets of X if C satisfies the following axioms : (OSCT1 ) C(∅) = C(X) = 1. (OSCT2 ) C(A ∪ B) ≥ C(A) ∧ C(B), ∀A, B ∈ 2X . \ ^ (OSCT3 ) C( Aα ) ≥ C(Aα ), ∀{Aα } ⊂ 2X . α∈Γ α∈Γ The pair (X, C) is called an ordinary smooth cotopological space (in short, oscts). We will denote the set of all osct’s on X as OSCT(X). Remark 2.5. If I = 2, then Definition 2.4 also coincides with the known definition of classical topology. Result 2.A [10, Proposition 2.7]. Let X be a nonempty set. We define two mappings f : OST(X) → OSCT(X) and g : OSCT(X) → OST(X) as follows, respectively : [f (τ )](A) = τ (Ac ), ∀τ ∈ OST(X), ∀A ∈ 2X and [g(C)](A) = C(Ac ), ∀C ∈ OSCT(X), ∀A ∈ 2X . Then f and g are well-defined. Furthermore g ◦ f = idOST(X) and f ◦ g = idOSCT(X) . Remark 2.6. Let f (τ ) = Cτ and g(C) = τC . Then, by Result 2.A, we can easily see that τCτ = τ and CτC = C. (c) 0 [τ ]∗r [ [τ ]∗s , ∀r ∈ I1 , where I1 = [0, 1) and I0 = s>r (0, 1]. 3. Closures and Interiors in Ordinary Smooth Topological Spaces Definition 3.1. Let (X, τ ) be an ordinary smooth topological space and let A ∈ 2X . Then the ordinary smooth closure of A in (X, τ ), denoted by Ā, is defined by if Cτ (A) = 1, 0, T X Ā = {F ∈ 2 : A ⊂ F and Cτ (A) < Cτ (F)}, if Cτ (A) 6= 1. Proposition 3.2. Let (X, τ ) be an ordinary smooth topological space and let A, B ∈ 2X . Then : (a) Cτ (A) ≤ Cτ (Ā). (b) If B ⊂ A and Cτ (B) ≤ Cτ (A), then B̄ ⊂ Ā. Proof. (a) From Definition 3.1 and the condition (OSCT1 ), it is obvious. (b) Let A, B ∈ 2X . Suppose B ⊂ A and Cτ (B) ≤ Cτ (A). Case (i): Suppose Cτ (B) = 1. Then B̄ = B. Since Cτ (B) ≤ Cτ (A), Cτ (A) = 1. Thus Ā = A. Since B ⊂ A, B̄ ⊂ Ā. Case (ii): Suppose Cτ (B) 6= 1 and Cτ (A) = 1. Then Ā = A and B̄ = www.ijfis.org = \ {F ∈ 2X : B ⊂ F and Cτ (B) < Cτ (F)}. Closure, Interior and Compactness in Ordinary Smooth Topological Spaces | 232 International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 3, September 2014 Since Cτ (A) = 1, A ∈ {F ∈ 2X : B ⊂ F Cτ (F)}. Thus B̄ = \ X {F ∈ 2 :B⊂F and Cτ (B) < and Cτ (B) < Cτ (F)} ⊂ A. So B̄ ⊂ Ā. Case (ii): Cτ (B) 6= 1 and Cτ (A) 6= 1. Then B̄ = \ {F ∈ 2X : B ⊂ F and Cτ (B) < Cτ (F)} B) 6= 1. Then A∪B \ = {F ∈ 2X : A ∪ B ⊂ F and Cτ (A ∪ B) < Cτ (F)} \ ⊃ {F ∈ 2X : A ∪ B ⊂ F and Cτ (A) ∧ Cτ (B) < Cτ (F)} [By the condition (OSCT2)] \ = {F ∈ 2X : A ⊂ F, B ⊂ F and Cτ (A) < Cτ (F) or Cτ (B) < Cτ (F )} \ = {F ∈ 2X : (A ⊂ F, B ⊂ F and Cτ (A) < Cτ (F)) or and Ā = \ {F ∈ 2X : A ⊂ F and Cτ (A) < Cτ (F)}. Thus, by the hypothesis, V̄ ⊂ Ā. This completes the proof. Proposition 3.3. Let (X, τ ) be an ordinary smooth topological space, and let A, B ∈ 2X . Then: (a) ¯ ∅ = ∅. (b) A ⊂ Ā. (c) Ā ⊂ Ā¯. (A ⊂ F, B ⊂ F and Cτ (B) < Cτ (F))} \ = [{F ∈ 2X : A ⊂ F and Cτ (A) < Cτ (F)} ∪ {F ∈ 2X : B ⊂ F and Cτ (B) < Cτ (F)}] \ ⊃ [ {F ∈ 2X : A ⊂ F and Cτ (A) < Cτ (F)}] ∩ [{F ∈ 2X : B ⊂ F and Cτ (B) < Cτ (F)}] = Ā ∩ B̄. Hence, in any cases, Ā ∩ B̄ ⊂ A ∪ B. (d) Ā ∩ B̄ ⊂ A ∪ B. Proof. (a) From Definition 3.1, it is obvious. (b) Let A ∈ 2X . Case (i): Suppose Cτ (A) = 1. Then A = Ā. Case (ii): Suppose Cτ (A) 6= 1. Then, by Definition 3.1, A ⊂ Ā. Thus A ⊂ Ā. (c) By (b), A ⊂ Ā. Moreover, Cτ (A) ≤ Cτ (Ā), by Proposition 3.2 (a). Thus, by Proposition 3.2 (b), Ā ⊂ Ā¯. (d) Let A, B ∈ 2X . Case (i): Suppose Cτ (A) = Cτ (B) = 1. Then, by the condition (OSCT2 ), Cτ (A ∪ B) = 1. Thus A ∪ B = A ∪ B. By the hypothesis, Ā = A and B̄ = B. So Ā ∩ B̄ ⊂ A ∪ B. Case (ii): Suppose Cτ (A) = 1, Cτ (B) 6= 1 and Cτ (A ∪ B) 6= 1. Then, by the condition (OSCT2 ), Cτ (A ∪ B) ≥ Cτ (B). Thus, by Proposition 3.2 (b), B̄ ⊂ A ∪ B. So Ā ∩ B̄ ⊂ A ∪ B. Case (iii): Suppose Cτ (A) = 1, Cτ (B) 6= 1 and Cτ (A∪B) = 1. Then Ā ⊂ A ∪ B. Thus Ā ∩ B̄ ⊂ A ∪ B. Case (iv): Suppose Cτ (A) 6= 1, Cτ (B) = 1 and Cτ (A∪B) 6= 1. Then it is similar to Case (ii). Case (v): Suppose Cτ (A) 6= 1, Cτ (B) = 1 and Cτ (A ∪ B) = 1. Then it is similar to Case (iii). Case (vi): Suppose Cτ (A) 6= 1, Cτ (B) 6= 1 and Cτ (A∪B) = 1. Then, by Proposition 3.2 (b), Ā ⊂ A ∪ B and B̄ ⊂ A ∪ B. Thus Ā ∩ B̄ ⊂ A ∪ B. Case (vii): Suppose Cτ (A) 6= 1, Cτ (B) 6= 1 and Cτ (A ∪ 233 | Jeong Gon Lee, Kul Hur, and Pyung Ki Lim Definition 3.4 Let (X, τ ) be an ordinary smooth topological space, let r ∈ I and let A ∈ 2X . Then the [τ ]r - closure of A, denoted by clr (A), is defined by clr (A) = \ {F ∈ 2X : F ∈ F[τ ]r and A ⊂ F}, where F[τ ]r denotes the set of all [τ ]r -closed sets in X. Remark 3.5. Let (X, τ ) be a classical topological space. Then: (a) T can be identified with an ordinary smooth topology τT on X defined as follows : τT : 2X → I is the mapping given by for each A ∈ 2X τT (A) = 1, if 0, otherwise. A ∈ T, ∗ In fact, [τT ]0 = T . (b) Also T can be identified with an ordinary smooth cotopology CT on X defined as follows : CT : 2X → I is the mapping given by for each A ∈ 2X , CT (A) = 1, if 0, otherwise. Ac ∈ T, (c) We can calculate the ordinary smooth closure of A w.r.t http://dx.doi.org/10.5391/IJFIS.2014.14.3.231 τT , denoted by clτT (A), defined by Definition 3.1: = \ [ \ {F ∈ 2X : A ⊂ F and Cτ (F) ≥ r}] r>Cτ (A) A, T clτT (A) = {F ∈ 2X : A ⊂ F and CτT (A) < CτT (F)} if CτT (A) = 1, = Ā. if CτT (A) 6= 1. Proposition 3.6. Let (X, τ ) be an ordinary smooth topological space and let A ∈ 2X . (a) cl[τ ]r (A) = clr (A), ∀r ∈ I0 . (b) If Cτ (A) = 1, then Ā = clr (A), ∀r ∈ I0 . T (c) If Cτ (A) 6= 1, then Ā = r>Fτ (A) clr (A). Proof. (a) From Result 2.B and Remark 3.5 (a), it is obvious that τ[τ ]r ∈ OST(X) for each r ∈ I0 . Case (i) : Suppose Cτ[τ ]r (A) = 1. Then, by Remark 3.5(b), T A ∈ [τ ]r . Thus A ∈ F[τ ]r . So clr (A) = {F ∈ 2X : F ∈ F[τ ]r and A ⊂ F} = A. On the other hand, by the hypothesis and Remark 3.5 (c), clτ[τ ]r (A) = A. Hence clτ[τ ]r (A) = clr (A). c Case (ii): Suppose Cτ[τ ]r (A) 6= 1. Then clτ[τ ]r (A) \ = {F ∈ 2X : A ⊂ F and Cτ[τ ]r (A) < Cτ[τ ]r (F)} \ = {F ∈ 2X : A ⊂ F and Cτ[τ ]r (F) = 1} [By Remark 3.5(b)] \ = {F ∈ 2X : A ⊂ F and F ∈ F[τ ]r } [By Remark 3.5(b)] = clr (A). [By Definition 3.4] (b) Suppose Cτ (A) = 1. Then, by Definition 3.1, Ā = A. Let r ∈ I0 . Then, by the hypothesis, τ (Ac ) = Cτ (A) = 1 ≥ r. Thus Ac ∈ [τ ]r . So A ∈ Fτ[τ ]r . Hence, by Definition 3.4, clr (A) = A. Therefore Ā = clr (A) for each r ∈ I0 . (c) Suppose Cτ (A) 6= 1. Then \ \ \ [ {F ∈ 2X : F ∈ F[τ ]r and A ⊂ F}] r>Cτ (A) = \ \ [ {F ∈ 2X : A ⊂ F and Fc ∈ [τ ]r }] r>Cτ (A) = [By Definition 3.1] This completes the proof. Definition 3.7. Let (X, τ ) be an ordinary smooth topological space and let A ∈ 2X . Then the ordinary smooth interior of A ◦ in (X, τ ),denoted by A, is defined as follows: A, ◦ S A= {U ∈ 2X : U ⊂ A and τ (U) > τ (A)} if τ (A) = 1, if τ (A) 6= 1. Proposition 3.8. Let (X, τ ) be an ordinary smooth topological space and let A, B ∈ 2X . Then: ◦ (a) τ (A) ≤ τ (A). ◦ ◦ (b) If B ⊂ A and τ (B) > τ (A), then B ⊂ A. Proof. (a) from Definition 3.7 and the condition (OST1 ), it is obvious. (b) The proof is similar to that of Proposition 3.2 (b). Proposition 3.9. Let (X, τ ) be an ordinary smooth topological space and let A, B ∈ 2X . Then: ◦ (a) X = X. ◦ (b) A ⊂ A. ◦ ◦ (c) (A)◦ ⊂ A. ◦ ◦ (d) (A ∩ B)◦ ⊂ A ∪ B. Proof. The proofs are similar to these of Proposition 3.3. Definition 3.10. Let (X, τ ) be an ordinary smooth topological space, let r ∈ I and let A ∈ 2X . Then the [τ ]r -interior of A, denoted by intr (A), is defined by intr (A) = [ {U ∈ 2X : U ∈ [τ ]r and U ⊂ A}. Proposition 3.11. Let (X, τ ) be an ordinary smooth topological space and let A ∈ 2X . (a) intτ[τ ]r (A) = intr (A), ∀r ∈ I0 , where intτ[τ ]r (A) is the ordinary smooth interior of A in (X, τ[τ ]r ), defined by clr (A) r>Cτ (A) = \ = {F ∈ 2X : A ⊂ F and Cτ (A) < Cτ (F)} \ \ [ {F ∈ 2X : A ⊂ F and Cτ (F) = τ (Fc ) ≥ r}] r>Cτ (A) www.ijfis.org A, S intτ[τ ]r (A) = {U ∈ 2X : U ⊂ A and τ[τ ]r (U) > τ[τ ]r (A)} if τ[τ ]r (A) = 1, if τ[τ ]r (A). ◦ (b) If τ (A) = 1, then A = intr (A), ∀r ∈ I0 . ◦ S (c) If τ (A) 6= 1, then A = r>τ (A) intr (A). Closure, Interior and Compactness in Ordinary Smooth Topological Spaces | 234 International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 3, September 2014 Proof. The proofs are similar to these of Proposition 3.6. 4. Ordinary Smooth Open Preserving Mappings Definition 4.1. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological spaces. Then a mapping f : X → Y is said to be ordinary smooth preserving[resp. ordinary strict smooth preserving] if for any A, B ∈ 2Y , Definition 4.3. Proposition 4.5. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological space. Suppose f : X → Y is an injective, ordinary strict smooth preserving and ordinary smooth continuous mapping. Then f (Ā) ⊂ f (A), ∀A ∈ 2X . Proof. Let A ∈ 2X . Case (i): Suppose Cτ2 (f (A)) = 1. Then 1 = Cτ2 (f (A)) ≤ Cτ1 (f −1 (f (A))) τ2 (B) ≤ τ2 (A) ⇔ τ1 (f −1 (B)) ≤ τ1 (f −1 (A)) = Cτ1 (A). [resp.τ2 (B) < τ2 (A) ⇔ τ1 (f −1 (B)) < τ1 (f −1 (A))]. The following is the characterization of Definition 4.1. Theorem 4.2. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological spaces: Suppose f : X → Y is an ordinary smooth preserving [resp. an ordinary strict smooth preserving] mapping. Then for any A, B ∈ 2Y , Cτ2 (B) ≤ Cτ2 (A) ⇔ Cτ1 (f −1 (B)) ≤ Cτ1 (f −1 (A)) [resp. Cτ2 (B) < Cτ2 (A) ⇔ Cτ1 (f −1 (B)) < Cτ1 (f −1 (A))]. Proof. (i) Suppose f is ordinary smooth preserving and let A, B ∈ 2X . Then Cτ2 (B) ≤ Cτ2 (A) c [Since f is injective] Thus Cτ1 (A) = 1. So, by Definition 3.1, Ā = A. Hence, by the hypothesis, f (A) = f (A) = f (Ā) Case (ii): Suppose Cτ2 (f (A)) 6= 1 and Cτ1 (A) = 1. Then clearly Ā = A. Thus, by Proposition 3.3 (b), f (Ā) = f (A) ⊂ f (A). Case (iii): Suppose Cτ2 (f (A)) 6= 1 and Cτ1 (A) 6= 1. Then f −1 (f (A)) \ = f −1 [ {F ∈ 2Y : f (A) ⊂ F ⊂f −1 and Cτ2 (f(A)) < Cτ2 (F)}] [By Definition 3.1] \ [ {F ∈ 2Y : A ⊂ f −1 (F ) c ⇔τ2 (B ) ≤ τ2 (A ) [By the definition of Cτ for each τ ∈ OST(X)] ⇔τ1 (f −1 (B c )) ≤ τ1 (f −1 (Ac )) [By the hypothesis] ⇔τ1 ((f −1 (B))c ) ≤ τ1 ((f −1 (A))c ) and Cτ1 (A) < Cτ1 (f −1 (F))] [Since f is injective and ordinary smooth preserving] \ = {f −1 (F ) ∈ 2X : A ⊂ f −1 (F ) and Cτ1 (A) < Cτ1 (f −1 (F))} ⇔Cτ1 (f −1 (B)) ≤ Cτ1 (f −1 (A)). = \ {B ∈ 2X : A ⊂ B (ii) Suppose f is ordinary strict smooth preserving. Then the proof is similar to that of (i). and Cτ1 (A) < Cτ1 (B)} = Ā Definition 4.3 [9]. Let (X, τ1 ) and (Y, τ2 ) be two ordinary Hence, in any cases smooth topological spaces. Then a mapping f : X → Y is said to be ordinary smooth continuous if τ2 (A) ≤ τ1 (f −1 (A)), ∀A ∈ f (A) ⊂ f (A), 2Y . Theorem 4.4. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological spaces and let f : X → Y be a mapping. Then f is ordinary smooth continuous if and only if Cτ2 (A) ≤ Cτ1 (f −1 (A)), ∀A ∈ 2Y . The following is the immediate result of Result 2.A and 235 | Jeong Gon Lee, Kul Hur, and Pyung Ki Lim ∀A ∈ 2X . Proposition 4.6. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological space. Suppose f : X → Y is an ordinary strict smooth preserving and ordinary smooth continuous mapping. Then f −1 (A) ⊂ f −1 (Ā), ∀A ∈ 2Y . Proof. Let A ∈ 2Y . Case (i): Suppose Cτ2 (A) = 1. Since f is ordinary smooth continuous, Cτ1 (f −1 (A)) = 1. Then f −1 (A) = f −1 (A). By the hypothesis, A = Ā. Thus f −1 (A) = f −1 (Ā). So http://dx.doi.org/10.5391/IJFIS.2014.14.3.231 f −1 (A) = f −1 (Ā). Case (ii): Suppose Cτ2 (A) 6= 1 and Cτ1 (f −1 (A)) = 1. By Proposition 3.3 (b), it is clear that A ⊂ Ā. Then f −1 (A) ⊂ f −1 (Ā). Since Cτ1 (f −1 (A)) = 1, f −1 (A) = f −1 (A). Thus f −1 (A) ⊂ f −1 (Ā). Case (iii): Suppose Cτ2 (A) 6= 1 and Cτ1 (f −1 For each i = 1, 2, 3, we define a mapping τi : 2X → I as follows : For each C ∈ 2X , τi (C) = 1, (A)) 6= 1. Then f −1 (Ā) \ = f −1 ( {F ∈ 2Y : A ⊂ F and Cτ2 (A) < Cτ2 (F)}) \ ⊃ f −1 ( {F ∈ 2Y : f −1 (A) ⊂ f −1 (F ) and Cτ1 (f −1 (A)) < Cτ1 (f −1 (F))}) [Since f is ordinary strict smooth preserving] = \ \ or C = X, τ1 (A) = 0.15, τ1 (B) = 0.80, τ2 (A) = 0.30, τ2 (B) = 0.50, τ3 (A) = 0.90, τ3 (B) = 0.80, τi (C) = 0.10, if C∈ / {∅, X, A, B}. Then it is obvious that τi ∈ OST(X) for each i = 1, 2, 3. Moreover, we can easily see that the identity mapping id : (X, τ1 ) → (X, τ2 ) is ordinary smooth open preserving but not ordinary smooth open. However, the identity mapping id : (X, τ2 ) → (X, τ3 ) is ordinary smooth open but not ordinary smooth open preserving. {f −1 (F ) ∈ 2X : f −1 (A) ⊂ f −1 (F ) and Cτ1 (f −1 (A)) < Cτ1 (f −1 (F))} = if C = ∅ {B ∈ 2X : f −1 (A) ⊂ B and Cτ1 (f −1 (A)) < Cτ1 (B)} = f −1 (A). Hence, in any cases, f −1 (A) ⊂ f −1 (Ā), ∀A ∈ 2Y . Proposition 4.10. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological spaces. Suppose f : X → Y is an ordinary strict smooth open preserving and ordinary smooth open ◦ mapping. Then f (A) ⊂ (f (A))◦ , ∀A ∈ 2X . Proof. Let A ∈ 2X . Case (i): Suppose τ1 (A) = 1. Since f is ordinary smooth open, τ2 (f (A)) = 1. Then (f (A))◦ = f (A). Since τ1 (A) = ◦ Definition 4.7 [9]. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological spaces. Then a mapping f : X → Y is said to be ordinary smooth open [resp. closed] if for each A ∈ 2X , τ1 (A) ≤ τ2 (f (A)) [resp.C1 (A) ≤ C2 (f(A))], ◦ 1, A = A. Thus (f (A))◦ = f (A). Case (ii): Suppose τ1 (A) 6= 1 and τ2 (f (A)) = 1. By ◦ ◦ Proposition 3.9 (b), A ⊂ A. Thus f (A) ⊂ f (A). Since ◦ τ2 (f (A)) = 1, (f (A))◦ = f (A). Thus f (A) ⊂ (f (A))◦ . Case (iii): Suppose τ1 (A) 6= 1 and τ2 (f (A)) 6= 1. Then ◦ where (X, C1 ) and (Y, C2 ) are ordinary smooth cotopological spaces. Definition 4.8. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological spaces. Then a mapping f : X → Y is said to be ordinary smooth open preserving [resp. ordinary strict smooth open preserving] if for any A, B ∈ 2X , f (A) = f [∪{U ∈ 2X : U ⊂ A and τ1 (U) > τ1 (A)}] ⊂ f [∪{U ∈ 2X : f (U ) ⊂ f (A) and τ2 (f(U)) > τ2 (f(A))}] [Since f is ordinary strict smooth open preserving] = ∪{f (U ) ∈ 2Y : f (U ) ⊂ f (A) and τ2 (f(U)) > τ2 (f(A))} = ∪{V ∈ 2Y : V ⊂ f (A) τ1 (B) ≤ τ1 (A) ⇒ τ2 (f (B)) ≤ τ2 (f (A)) [resp. and τ2 (V) > τ2 (f(A))} τ1 (B) < τ1 (A) ⇒ τ2 (f (B)) < τ2 (f (A))]. Notice that the concept of an ordinary smooth open preserving mapping differs from the concept of an ordinary smooth open mapping. Example 4.9. Let X = {a, b}, let A = {a} and let B = {b}. www.ijfis.org = (f (A))◦ . ◦ Hence, in any cases, f (A) ⊂ (f (A))◦ . Proposition 4.11. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological spaces. Suppose f : X → Y is an ordinary Closure, Interior and Compactness in Ordinary Smooth Topological Spaces | 236 International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 3, September 2014 strict smooth preserving and ordinary smooth continuous. Then Y −1 ◦ −1 for each A ∈ 2 , f (A) ⊂ (f (A)) . Proof. By using Definition 3.7, the proof is similar to that of Proposition 4.5. 5. is said to be ordinary smooth regular if for each A ∈ S(τ ), ◦ Some Types of Ordinary Smooth Compactness For an ordinary smooth topological spaces (X, τ ), let us define S(τ ) = {A ∈ 2X : τ (A) > 0} and S(τ ) will be called the support of τ . Definition 5.1. An ordinary smooth topological space (X, τ ) is said to be: (i) ordinary smooth compact if for every family {Aα }α∈Γ in S(τ ) covering X, there is a finite subset Γ0 of Γ such that S α∈Γ0 Aα = X. (ii) ordinary smooth almost compact if for every family {Aα }α∈Γ in S(τ ) covering X, there is a finite subset Γ0 of S Γ such that α∈Γ0 Āα = X. (iii) ordinary smooth nearly compact if for every family {Aα }α∈Γ in S(τ ) covering X, there is a finite subset Γ0 of Γ S such that α∈Γ0 (Āα )◦ = X, or equivalently for every family {Aα }α∈Γ in {A ∈ 2X : τ (A) > 0 and A = (Ā)◦ }, there S is a finite subset Γ0 of Γ such that α∈Γ0 Aα = X. The following is the characterization of Definition 5.1 (i). Theorem 5.2. Let (X, τ ) be an ordinary smooth topological space. Then (X, τ ) is ordinary smooth compact if and only if every family in S(τ ) having the finite intersection property (in short, F.I.P.) has a nonempty intersection. Proof. (⇒): Suppose (X, τ ) is ordinary smooth compact. Let {Aα }α∈Γ be the family in S(Cτ ) having the F.I.P., i.e., for any fiT T nite subset Γ0 ⊂ Γ, α∈Γ0 Aα 6= ∅. Assume that α∈Γ Aα = S ∅. Then clearly α∈Γ Aα c = X. Since Aα ∈ S(Cτ ) for each α ∈ Γ, Cτ (Aα ) = τ (Aα c ) > 0 for each α ∈ Γ. Thus {Aα c }α∈Γ is a covering of X. So, by the hypothesis, there S is a finite subset Γ0 ⊂ Γ such that α∈Γ0 Aα c = X. Hence T α∈Γ0 Aα = ∅. This is a contradiction. (⇐): Suppose the necessary condition holds. Let {Aα }α∈Γ S be a family in S(τ ) covering X. Then α∈Γ Aα = X. AsS sume that for any subset Γ0 ⊂ Γ, α∈Γ0 Aα 6= X. Then T c c α∈Γ0 Aα 6= ∅. Since {Aα }α∈Γ is a family in S(τ ), Cτ (Aα ) = τ (Aα ) > 0. Thus {Aα c }α∈Γ is the family in S(Cτ ) havT ing the F.I.P. So, by the hypothesis, α∈Γ Aα c 6= ∅. Hence S α∈Γ Aα 6= X. This is a contradiction. [ {B ∈ 2X : τ (A) ≤ τ (B) A= Proposition 5.4. Let (X, τ ) be an ordinary smooth topological space. (a) If (X, τ ) is ordinary smooth almost compact and ordinary smooth regular, then so is it. (b) If (X, τ ) is ordinary smooth nearly compact and ordinary smooth regular, then it is ordinary smooth compact. Proof. (a) Suppose (X, τ ) is ordinary smooth almost compact and ordinary smooth regular. Let {Aα }α∈Γ be a family in S(τ ) S covering X, i.e., α∈Γ Aα = X. Since (X, τ ) is ordinary smooth regular, for each α ∈ Γ, Aα = [ {Bα ∈ 2X : τ (Aα ) ≤ τ (Bα ) 237 | Jeong Gon Lee, Kul Hur, and Pyung Ki Lim and B̄α ⊂ Aα }. S Then clearly α∈Γ Bα = X and {Bα }α∈Γ is a family in S(τ ). Since (X, τ ) is ordinary smooth almost compact, there is a finite S subset Γ0 ⊂ Γ such that α∈Γ0 B̄α = X. Since B̄α ⊂ Aα S for each α ∈ Γ, α∈Γ0 Aα = X. Hence (X, τ ) is ordinary smooth compact. (b) The proof is quite to that of (a) taking into account that the ordinary smooth interior of an ordinary subset remains always smaller then itself. Proposition 5.5. Let (X, τ1 ) and (Y, τ2 ) be two ordinary smooth topological spaces and let f : X → Y be a surjective ordinary smooth continuous and ordinary strict smooth preserving mapping. (a) If (X, τ1 ) is ordinary smooth almost compact, then so is (Y, τ2 ). (b) If (X, τ1 ) is ordinary smooth nearly compact, then (Y, τ2 ) is ordinary smooth almost compact. Proof. (a) Let {Aα }α∈Γ be a family in S(τ2 ) covering Y , i.e., S α∈Γ Aα = Y . Since f is ordinary smooth continuous, τ2 (Aα ) ≤ τ1 (f −1 (Aα )) for each α ∈ Γ. S S Since α∈Γ Aα = Y , α∈Γ f −1 (Aα ) = X. Since τ2 (Aα ) > 0 for each α ∈ Γ, τ1 (f −1 (Aα )) > 0 for each α ∈ Γ. Thus {f −1 (Aα )}α∈Γ is a family in S(τ ) covering X, since (X, τ1 ) is ordinary smooth almost compact, there is a finite subset S Γ0 ⊂ Γ such that α∈Γ0 f −1 (Aα ) = X. Since f is surjective, f( Definition 5.3. An ordinary smooth topological space (X, τ ) and B̄ ⊂ A} [ α∈Γ0 f −1 (Aα )) = [ α∈Γ0 f (f −1 (Aα )) = X. http://dx.doi.org/10.5391/IJFIS.2014.14.3.231 Since f is ordinary smooth continuous and ordinary strict smooth preserving, by Proposition 4.5, f −1 (Aα ) ⊂ f −1 (Āα ), for each α ∈ Γ. So f (f −1 (Aα )) ⊂ f (f −1 (Āα )) = Āα , for each α ∈ Γ. Hence S α∈Γ0 Āα = X. Therefore (Y, τ2 ) is ordinary smooth almost compact. (b) The proof is similar to that of (a). 6. Conclusions It is difficult to investigate the compactness using the notions of the closure and the interior introduced by Ying [8, 9]. To handle the difficulty, we introduced the new definitions of the closure and the interior different from the Ying’s definitions. We discussed the topological properties based on the definitions. The new definitions help to naturally study some compactness in an ordinary smooth topological space. Conflict of Interest No potential conflict of interest relevant to this article was reported. Acknowledgments This work was supported by the research grant of the Wonkwang University in 2014. References [1] R. Badard, “Smooth axiomatics,” in Proceedings of the Ist International Fuzzy Systems Association (IFSA) Congress, Palma de Mallorca, Spain, July, 1986. [2] C. L. Chang, “Fuzzy topological spaces,” Journal of Mathematical Analysis and Applications, vol. 24, no. 1, pp. 182-190, Oct. 1968. http://dx.doi.org/10.1016/ 0022-247X(68)90057-7 [3] R. N. Hazra, S. K. Samanta, and K. C. Chattopadhyay, “Fuzzy topology redefined,” Fuzzy Sets and Systems, vol. 45, no. 1, pp. 79-82, Jan. 1992. http://dx.doi.org/10.1016/ 0165-0114(92)90093-J [4] K. C. Chattopadhyay, R. N. Hazra, and S. K. Samanta, “Gradation of openness: fuzzy topology,” Fuzzy Sets and www.ijfis.org Systems, vol. 49, no. 2, pp. 237-242, Jul. 1992. http: //dx.doi.org/10.1016/0165-0114(92)90329-3 [5] M. Demirci, “Neighborhood structures of smooth topological spaces,” Fuzzy Sets and Systems, vol. 92, no. 1, pp. 123-128, Nov. 1997. http://dx.doi.org/10.1016/ S0165-0114(96)00132-7 [6] A. A. Ramadan, “Smooth topological spaces,” Fuzzy Sets and Systems, vol. 48, no. 3, pp. 371-375, Jun. 1992. http://dx.doi.org/10.1016/0165-0114(92)90352-5 [7] M. K. El Gayyar, E. E. Kerre, and A. A. Ramadan, “Almost compactness and near compactness in smooth topological spaces,” Fuzzy Sets and Systems, vol. 62, no. 2, pp. 193-202, Mar. 1994. http://dx.doi.org/10.1016/ 0165-0114(94)90059-0 [8] M. Ying, “A new approach for fuzzy topology (I),” Fuzzy Sets and Systems, vol. 39, no. 3, pp. 303-321, Feb. 1991. http://dx.doi.org/10.1016/0165-0114(91)90100-5 [9] M. Ying, “A new approach for fuzzy topology (II),” Fuzzy Sets and Systems, vol. 47, no. 2, pp. 221-232, Apr. 1992. http://dx.doi.org/10.1016/0165-0114(92)90181-3 [10] P. K. Lim, B. G. Ryoo, and K. Hur, “Ordinary smooth topological spaces,” International Journal of Fuzzy Logic and Intelligent Systems, vol. 12, no. 1, pp. 66-76, Mar. 2012. http://dx.doi.org/10.5391/IJFIS.2012.12.1.66 [11] M. S. Cheong, G. B. Chae, K. Hur, and S. M. Kim, “The lattice of ordinary smooth topologies,” Honam Mathematical Journal, vol. 33, no. 4, pp. 453-465, 2011. http://dx.doi.org/10.5831/HMJ.2011.33.4.453 Jeong Gon Lee received the Ph.D. degree in The Department of Mathematics Education from Korea National University of Education. He is currently assistant professor in Wonkwang University, Korea. His research interests are measure theory, operator theory, mathematics education, category theory, hyperspace, and topology. At present he has worked as one of managing editors in Annals of Fuzzy Mathematics and Informatics (AFMI). E-mail: [email protected] Closure, Interior and Compactness in Ordinary Smooth Topological Spaces | 238 International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 3, September 2014 Kul Hur received the Ph.D. degree in the Department of Mathematics from Yonsei University. He was a professor in Wonkwang University. His research interests are category theory, hyperspace and topology. He retired from Wonkwang University on February 2012. At present he has worked as one of editors-in-chief in Annals of Fuzzy Mathematics and Informatics (AFMI). E-mail: [email protected] 239 | Jeong Gon Lee, Kul Hur, and Pyung Ki Lim Pyung Ki Lim received the Ph.D. degree in the Department of Mathematics from Chonnam National University, Korea. He is currently professor in Wonkwang University. His research interests are category theory, hyperspace and topology. E-mail: [email protected]