CUBO A Mathematical Journal Vol.10, N o ¯ 01, (33–41). March 2008 On Some What Fuzzy Faintly Semicontinuous Functions G. Palanichetty and G. Balasubramanian Ramanujan Institute for Advanced Studies in Mathematics University of Madras, Department of Mathematics I.R.T. Polytechnic College, Krishnagiri – 635 108 Tamilnadu Chepauk, Chennai – 600 005,India email: gpalanichetty@yahoo.co.in ABSTRACT In this paper the concept of some what fuzzy faintly semicontinuous functions, some what fuzzy faintly semiopen functions,weakly some what fuzzy faintly semiopen functions are introduced. Some characterizations and interesting prop- erties of these functions are discussed. RESUMEN El concepto de funciones fuzzy débil semicontinuas, funciones fuzzy débil semi- abiertas son introducidas. Caracterizaciones e propriedades de este tipo de fun- ciones son discutidas. Key words and phrases: Some what fuzzy faintly semicontinuous functions, some what fuzzy faintly semiopen functions, fuzzy semidense set, fuzzy semiseparable space, fuzzy DS- space. Math. Subj. Class.: 54A40. 34 G. Palanichetty and G. Balasubramanian CUBO 10, 1 (2008) 1 Introduction The concept of fuzzy sets was introduced by Zadeh [8]. Based on the concept of fuzzy sets, C.L.Chang [4] introduced and developed the concept of fuzzy topological spaces. Since then various important notions in the classical topology such as continuous functions [4] have been extended to fuzzy topological spaces. The concept of some what continuous functions was introduced and studied by Karl R. Gentry and Hughes B.Hoyle [5] in general topological spaces and this concept was extended to fuzzy topological spaces in [6].. In [2] the concept of fuzzy faintly α-continuous func- tions was introduced and studied. Also some characterizations of fuzzy faintly continuous functions are given. The purpose of this paper is to introduce and study the concept of some what fuzzy faintly semicontinuous functions and it is infact a generalization of fuzzy faintly continuous functions introduced and studied in [2]. Section 2 deals with prelimi- naries. Section 3 deals with some what fuzzy faintly semicontinuous functions and some of its characterizations. Section 4 deals with some what fuzzy faintly semiopen functions and some of its basic properties and characterizations. 2 Preliminaries In this paper by (X, T ) we mean fuzzy topological space in the sense of [4]. Let λ be a fuzzy set. We define closure of λ = clλ = ∧{µ/µ ≥ λ, µ is fuzzy closed } and Interior of λ = intλ = ∨{σ/σ ≤ λ, σ is fuzzy open}.In [7] the semiinterior and semiclosure of a fuzzy set λ are defined as S-int(λ) = ∨ {µ/µ ≤ λ, µ is fuzzy semiopen} and S-cl(λ) = ∧{µ/µ ≥ λ, µ is fuzzy semiclosed} and also shown that S-cl(1 − λ) = 1−S-int(λ) and S-int(1 − λ) = 1−S- cl(λ) . A fuzzy set λ in (X, T ) is called proper if λ 6= 0 and λ 6= 1.A fuzzy point xp in X is a fuzzy set in X defined by xp (y) = { p, p ∈ (0, 1] for y = x, y ∈ X; 0, for y 6= x, y ∈ X. x and p are respectively called the support and the value of xp. A fuzzy point xp is said to be quasi-coincident [1] with α, denoted by xpqα , if and only if p > α ′ (x) or p + α(x) > 1 where α ′ denotes the complement of α.. A fuzzy subset λ in a fuzzy topological space X is said to be a q− neighbourhood for a fuzzy point xp if and only if there exists a fuzzy open subset β such that xpqβ ≤ λ. A fuzzy set λ is called fuzzy θ-open [1] if and only if intθ (λ) = λ where intθ (λ) = ∨{xp ∈ X/ for some open q- neighborhood β of xp, clβ ≤ λ} and clθ (λ) = ∧ {µ/µ > λ, µ is θ - closed}. λ is called θ-closed if clθ(λ) = λ. A fuzzy set λ CUBO 10, 1 (2008) On Some What Fuzzy Faintly Semicontinuous Functions 35 in a fuzzy topological space (X, T ) is called dense (θ- dense, semidense) if there exists no fuzzy closed set (θ- closed set, semiclosed set ) µ such that λ < µ < 1. A fuzzy set λ in (X, T ) is called fuzzy semiopen [3] if for some fuzzy open set v, v ≤ λ ≤ cl (v) , equivalently λ is called fuzzy semiopen if λ ≤ cl int(λ) .If λ and µ are any two fuzzy sets in X and Y respectively, we define λ × µ : X × Y → I as follows: (λ × µ) (x, y) = min (λ (x) , µ (y)) . A fuzzy topological space X is product related to a fuzzy topological space Y if for any fuzzy set v in X and ξ in Y whenever λ ′ (= 1 − λ) � v and µ′ (1 − µ) � ξ imply λ′ × 1 ∨ 1 × µ′ ≥ v × ξ, where λ is a fuzzy open set in X and µ is a fuzzy open set in Y,there exists a fuzzy open set λ1 and a fuzzy open set µ1in Y such that λ ′ 1 ≥ v or µ ′ 1 ≥ ξ and λ ′ 1 × 1 ∨ 1 × µ′1 = λ ′ × 1 ∨ 1 × µ′. If (X, T ) and (Y, S) are any two fuzzy topological spaces, we define a product fuzzy topology T × S on X × Y to be that fuzzy topology for which B = {λ × µ/λ ∈ T, µ ∈ S} forms a base. 3 Some what fuzzy faintly semicontinuous functions In [2] the concept of fuzzy faintly continuous function is given as follows : Definition 3.1 Let f : (X, T1) → (Y, T2) be a function from the fuzzy topological space (X, T1) to the fuzzy topological space (Y, T2). f is called fuzzy faintly continuous if f −1 (λ) is fuzzy open for every fuzzy θ− open set λ in Y . In [7] the concept of somewhat fuzzy semicontinuous function is introduced as follows. Definition 3.2 Let f : (X, T ) → (Y, S) be a function from the fuzzy topological space (X, T ) to the fuzzy topological space (Y, S). f is called somewhat fuzzy semicontinuous if λ ∈ S and f −1 (λ) 6= 0 implies there exists a fuzzy semiopen set µ of X such that µ ≤ f −1(λ). We are now ready to make the following: Definition 3.3 Let (X, T ) and (Y, S) be any two fuzzy topological spaces. A function f : (X, T ) → (Y, S) is called some what fuzzy faintly semicontinuous if for every fuzzy θ- open set λ in Y such that f −1(λ) 6= 0, there exists a fuzzy semiopen set 0 6= µ in (X, T ) such that µ ≤ f −1(λ). Clearly every fuzzy faintly continuous function [2] is some what fuzzy faintly semicon- tinuous but the converse is not true as the following example shows. 36 G. Palanichetty and G. Balasubramanian CUBO 10, 1 (2008) Example 3.4 Consider Example 2.3 [2]. Let X = Y = I = [0, 1]. Let µ1, µ2, µ3 be fuzzy sets on I defined as follows: µ1 (x) = { 0, 0 ≤ x ≤ 1 2 2x − 1, 1 2 ≤ x ≤ 1 µ2 (x) =    1, 0 ≤ x ≤ 1 4 −4x + 2, 1 4 ≤ x ≤ 1 2 0, 1 2 ≤ x ≤ 1 and µ3 (x) = { x, 0 ≤ x ≤ 1 4 ; 1, 1 4 ≤ x ≤ 1. Clearly T = {0, µ2, 1}, S = {0, µ1, µ2, µ1 ∨ µ2, 1} are two fuzzy topologies on I. Let f : (I, T ) → (I, S) be defined as follows f (x) = x, for each x in I. Now we can verify f −1 (µ3) = µ3 and f −1 (1) = 1 are fuzzy semiopen subsets of (I, T ) and also µ3 and 1 are the only fuzzy θ-open subsets of (I, S). It is easy to see that µ3 and 1 are fuzzy semiopen sets in (I, T ) . Hence f is somewhat fuzzy faintly semicontinuous; But since f −1(µ3) = µ3 6∈ T1, f is not fuzzy faintly continuous. Proposition 3.5 Let (X, T ) and (Y, S) be any two fuzzy topological spaces. Let f : (X, T ) → (Y, S) be a function.Then the following are equivalent. (a) f is some what fuzzy faintly semicontinuous. (b) If λ is a fuzzy θ- closed set of Y such that f −1(λ) 6= 1, then there exists a proper fuzzy semiclosed set µ of X such that µ ≥ f −1(λ). (c) If λ is a fuzzy semidense set, then f (λ) is a fuzzy θ - dense set in Y . Proof. (a)⇒ (b) Suppose f is some what fuzzy faintly semicontinuous and λ is any fuzzy θ-closed set in Y such that f −1 (λ) 6= 1. Therefore clearly 1 − λ is fuzzy θ - open and f −1 (1−λ) = 1−f −1(λ) 6= 0. But by (a) there exists a fuzzy semiopen set µ∗ in (X, T ) such that µ ∗ 6= 0 and µ∗ ≤ f −1(1 − λ). Therefore 1 − µ∗ ≥ 1 − f −1 (1 − λ) = 1 − [ 1 − f −1 (λ) ] = f −1 (λ) . Put 1 − µ∗ = µ. Clearly µ is proper fuzzy semiclosed set and µ ≥ f −1(λ). This shows (a) ⇒ (b). (b)⇒ (c) Let λ be a fuzzy semidense set in X and suppose f (λ) is not fuzzy θ - dense in Y . Then there exists a fuzzy θ- closed set say µ ∗ such that f (λ) < µ ∗ < 1. (A) Now µ ∗ < 1 ⇒ f −1(µ∗) 6= 1. Then by (b) there exists a proper fuzzy semiclosed set σ in (X, T ) such that σ ≥ f −1(µ∗). But by (A), f −1(µ∗) > f −1[f (λ)] ≥ λ that is σ > λ. This CUBO 10, 1 (2008) On Some What Fuzzy Faintly Semicontinuous Functions 37 implies there exists a proper fuzzy semiclosed set σ such that σ > λ which is a contradiction, since λ is fuzzy semidense set. This proves (b) ⇒ (c). (c) ⇒ (a) Let λ be any fuzzy θ-open set in (Y, S) and suppose f −1(λ) 6= 0 and hence λ 6= 0. We want to show that f is some what fuzzy faintly semicontinuous. That is we want to show that there exists a fuzzy semiopen set µ in (X, T ) such that 0 6= µ ≤ f −1(λ).That is we want to show that S-int[f −1 (λ)] 6= 0.Suppose S-int[f −1(λ)] = 0.Then S-cl [ 1 − f −1(λ) ] = 1 − S-int[f −1(λ)] = 1 − 0 = 1. This means 1−f −1(λ) is fuzzy semidense in X. Now by (c), f [1−f −1(λ)] is fuzzy θ-dense in Y. That is clθf [1 − f −1 (λ)] = 1; but f [1 − f −1(λ)] = f [f −1(1 − λ)] ≤ 1 − λ < 1 (since λ 6= 0). Since 1−λ is fuzzy θ-closed and f [1−f −1(λ)] ≤ 1−λ,clθ f [1−f −1 (λ)] ≤ 1−λ.That is 1≤1−λ ⇒ λ≤0 ⇒ λ = 0, which is a contradiction to the fact that λ 6= 0. Therefore we must have S-int[f −1 (λ)] 6= 0. This proves that f is some what fuzzy faintly semicontinuous. 2 Proposition 3.6 Let (X, T ) and (Y, S) be fuzzy topological spaces and f : (X, T ) → (Y, S) be some what fuzzy faintly semicontinuous. Let A be a subset of X such that χA ∧ µ 6= 0 for some 0 6= µ in (X, T ) . Let T /A be the induced fuzzy topology on A. Then f /A : (A, T /A) → (Y, S) is some what fuzzy faintly semicontinuous. Proof. Suppose λ is a fuzzy θ - open set in (Y, S) such that f −1(λ) 6= 0. Since f is some what fuzzy faintly semicontinuous, there exists a fuzzy semiopen set µ in (X, T ) such that µ 6= 0 and µ ≤ f −1(λ). But µ/A ∈ T /A and µ/A 6= 0 (since χA ∧ µ 6= 0 for all µ ∈ T ). Now (f /A) −1 (λ) (x) = λ [(f /A) (x)] = λf (x) > µ (x) = µ/A (x) for x ∈ A. Also µ is a fuzzy semiopen set in (X, T ) implies µ/A is fuzzy semiopen in (A, T /A) and µ/A < (f /A) −1 (λ). This proves that f /A is some what fuzzy faintly semicontinuous. 2 Definition 3.7 Let X be a set, T and T ′ be any two fuzzy topologies for X. We say that T ′ is weakly semiequivalent to T if λ 6= 0 is a fuzzy semiopen set in (X, T ) , then there is a fuzzy semiopen set µ in (X, T ′) such that µ 6= 0 and µ ≤ λ. Proposition 3.8 Let (X, T ) and (Y, S) be any two fuzzy topological spaces and suppose that f : (X, T ) → (Y, S) is some what fuzzy faintly semicontinuous. Let T ′ be a fuzzy topology weakly semiequivalent to T . Then f : (X, T ′) → (Y, S) is some what fuzzy faintly semicontinuous. Proof. Let λ be a fuzzy θ- open set in S such that f −1 (λ) 6= 0. Since f : (X, T ) → (Y, S) is some what fuzzy faintly semicontinuous , there is a fuzzy semiopen set µ in (X, T ) and µ 6= 0 38 G. Palanichetty and G. Balasubramanian CUBO 10, 1 (2008) such that µ ≤ f −1(λ). But by the hypothesis T ′ is weakly semiequivalent to T. Therefore there exists a fuzzy semiopen set µ ∗ such that µ ∗ in (X, T ′ ) and µ ∗ 6= 0 and µ∗ < µ. But µ < f −1 (λ) implies µ ∗ ≤ f −1(λ). This means f : (X, T ′) → (Y, S) is some what fuzzy faintly semicontinuous. 2 Definition 3.9 Let Y be a set. Let S and S′ be any two fuzzy topologies in Y. We say that S ′ is weakly θ-equivalent to S if λ 6= 0 is a fuzzy θ-open set in (Y, S′) , then there exists a fuzzy θ-open set µ in (Y, S) such that µ ≤ λ. Proposition 3.10 Let (X, T ) and (Y, S) be any two fuzzy topological spaces and suppose f : (X, T ) → (Y, S) is some what fuzzy faintly semicontinuous. Let S′ be a fuzzy topology weakly θ-equivalent to S. Then f : (X, T ) → (Y, S′) is some what fuzzy faintly semicontinuous. Proof. Let λ be a fuzzy θ-open set in (Y, S′) such that f −1 (λ) 6= 0. Since S′ is weakly θ-equivalent to S, there exists a fuzzy θ-open set λ ∗ in (Y, S) such that 0 6= λ∗ ≤ λ. Now 0 6= f −1 (λ∗) ≤ f −1 (λ) .Since f is some what fuzzy faintly semicontinuous from (X, T ) to (Y, S) there exists a fuzzy semiopen set in (X, T ) say µ such that µ 6= 0 and µ ≤ f −1 (λ∗) . This means µ ≤ f −1 (λ) and so f is some what fuzzy faintly semicontinuous from (X, T ) to (Y, S ′ ). 2 Proposition 3.11 Let (X, T ) and (Y, S) be fuzzy topological spaces. Suppose that f : (X, T ) → (Y, S) is some what fuzzy faintly semicontinuous, T ′ and S′ are fuzzy topolo- gies for X and Y respectively such that T ′ is weakly semiequivalent to T and S′ is weakly θ-equivalent to S. Then f : (X, T ′) → (Y, S′) is some what fuzzy faintly semicontinuous. Proof. Proof follows from Propositions 3.8 and 3.10. 2 Definition 3.12 A fuzzy topological space (X, T ) is said to be fuzzy separable (semisepa- rable) if there exists a fuzzy dense (semidense) set λ in (X, T ) such that λ 6= 0 for atmost countably many points of X. Proposition 3.13 If f : (X, T ) → (Y, S) is a some what fuzzy faintly semicontinuous function and if X is fuzzy semiseparable, then Y is fuzzy θ-separable. Proof. Since X is fuzzy semiseparable, there exists a fuzzy dense set λ such that λ 6= 0 for atmost countably many points of X. Also since f is some what fuzzy faintly semicontinuous, it follows by Proposition 3.5 that f (λ) is fuzzy θ- dense in (Y, S) and since λ 6= 0 and λ is semidense for atmost countably many points, it follows that f (λ) 6= 0 for atmost countably many points. Thus we find that (Y, S) is fuzzy θ-separable. 2 CUBO 10, 1 (2008) On Some What Fuzzy Faintly Semicontinuous Functions 39 4 Some what fuzzy faintly semiopen functions Definition 4.1 Let (X, T ) and (Y, S) be fuzzy topological spaces. f : (X, T ) → (Y, S) is called some what fuzzy faintly semiopen function if and only if for any fuzzy semiopen set λ, λ 6= 0 in (X, T ) implies that there exists a fuzzy θ- open set µ in (Y, S) such that µ 6= 0 and µ < f (λ).That is intθ [f (λ)] 6= 0. Proposition 4.2 Let (X, T ), (Y, S) and (Z, R) be fuzzy topological spaces. Suppose that f : (X, T ) → (Y, S) and g : (Y, S) → (Z, R) are some what fuzzy faintly semiopen functions. Then g ◦ f : (X, T ) → (Z, R) is some what fuzzy faintly semiopen. Proof. Let λ be a fuzzy semiopen set in T. Since f is some what fuzzy faintly semiopen then there exists a fuzzy θ-open set µ in S such that µ ≤ f (λ). Now intθf (λ) ∈ S and since g is some what fuzzy faintly semiopen, then there exists a fuzzy θ-open set γ in (Z, R) such that γ < g [intf (λ)] . But g [intf (λ)] < g [f (λ)] . Thus we find that there exists a fuzzy θ-open set γ in (Z, R) such that γ < (g ◦ f ) (λ) . This proves g ◦ f is some what fuzzy faintly semiopen. 2 Proposition 4.3 Let (X, T ) and (Y, S) be fuzzy topological spaces. Let f : (X, T ) → (Y, S) be an onto function. Then the following are equivalent. (a) f is some what fuzzy faintly semiopen. (b) If λ is a fuzzy semiclosed set in X such that f (λ) 6= 1, then there exists a fuzzy θ-closed set µ in Y such that µ 6= 1 and µ > f (λ). Proof. (a) ⇒ (b). Let λ be a fuzzy semiclosed set in X such that f (λ) 6= 1. Then (1 − λ) is a fuzzy semiopen set such that f (1 − λ) = 1 − f (λ) 6= 0. Since f is some what fuzzy faintly semiopen, there exists a fuzzy θ-open set σ in S such that σ 6= 0 and σ ≤ f (1 − λ). Now 1 − σ is a fuzzy θ-closed set in Y such that 1 − σ 6= 1 and σ < f (1 − λ). Put 1 − σ = µ. Then µ > 1 − f (1 − λ) = f (λ). This proves (a) ⇒ (b). (b) ⇒ (a). Let λ in (X, T ) be a fuzzy semiopen set such that λ 6= 0. Then 1 − λ is fuzzy semiclosed and 1 − λ 6= 1, f (1 − λ) = 1 − f (λ) 6= 1.. Hence by (b) there exists a fuzzy θ-closed set µ in Y such that µ 6= 1 and µ > f (1 − λ) = 1 − f (λ), that is, f (λ) > 1 − µ and let 1 − µ = γ (say). Clearly γ is a fuzzy θ-open set in Y such that γ < f (λ) and γ 6= 0. Hence f is some what fuzzy faintly semiopen. This proves (b) ⇒ (a). 2 Proposition 4.4 Suppose (X, T ) and (Y, S) be fuzzy topological spaces. Let f : (X, T ) → (Y, S) be any onto function. Then the following are equivalent. 40 G. Palanichetty and G. Balasubramanian CUBO 10, 1 (2008) (a) f is some what fuzzy faintly semiopen (b) If λ is a fuzzy θ- dense set in Y, then f −1 (λ) is fuzzy semidense set in X. Proof. (a) ⇒ (b). Assume f is some what fuzzy faintly semiopen. Suppose λ is fuzzy θ-dense set in Y we have to show that f −1 (λ) is fuzzy semidense in X. Suppose not, then there exists a fuzzy semiclosed set µ in X such that f −1 (µ) < µ < 1. Now λ = f f −1 (λ) < f (µ) < f (1)(Since f is onto). Since f is some what fuzzy faintly semiopen by Proposition 4.3, there exists a fuzzy θ-closed set δ in Y such that f (µ) < δ.. Thus we find λ < f (µ) < δ < 1, which is a contradiction to our hypothesis that λ is fuzzy θ-dense in X. Hence f −1 (λ) must be fuzzy semidense set. This proves (a) ⇒ (b). (b) ⇒ (a). Assume that f −1 (λ) is fuzzy semidense in X where λ is fuzzy θ-dense in Y. We want to show that f is some what fuzzy faintly semiopen. Assume that λ ∈ T and λ 6= 0, be a fuzzy semiopen set in (X, T ) we have to show that intθf (λ) 6= 0. Suppose not, then intθf (λ) = 0 whenever λ ∈ T then clθ [1 − f (λ)] = 1−intθf (λ) = 1 − 0 = 1. That is, 1 − f (λ) is fuzzy θ-dense in Y. Therefore by assumption f −1 [1 − f (λ)] is fuzzy semidense in X. Therefore 1 =S-cl [ f −1 (1 − f (λ)) ] =S-cl[1 − λ] = 1 − λ. This shows λ = 0 which is a contradiction and so S-intf (λ) 6= 0.This proves (b) ⇒ (a). 2 In [6], the concept of fuzzy D-space is defined as follows: Definition 4.5 A fuzzy topological space (X, T ) is called a fuzzy D-space if every non-zero fuzzy open set λ of (X, T ) is dense in X. Now we are ready to make the following. Definition 4.6 A fuzzy topological space (X, T ) is called a fuzzy DS- space (Dθ- space) if every non-zero fuzzy semiopen set λ(non-zero fuzzy θ- open set) in X is semidense (θ-dense) in X. Proposition 4.7 Let f : (X, T ) → (Y, S) be some what fuzzy faintly semicontinuous. Sup- pose X is a fuzzy DS- space. Then Y is a fuzzy Dθ- space. Proof. Let λ be a non-zero fuzzy θ- open set in Y. We want to show that λ is fuzzy θ- dense inY. Suppose not. Then there exists a fuzzy θ- closed set µ in Y such that λ < µ < 1. Therefore f −1 (λ) < f −1 (µ) < f −1 (1) = 1.Since λ 6= 0, f −1 (λ) 6= 0 and since f is some what fuzzy faintly semicontinuous, 0 6= S-intf −1 (λ) < f −1 (µ) < S-clf −1 (µ) < 1. This is a contradiction to the assumption that (X, T ) is a fuzzy DS - space. Hence Y is a fuzzy Dθ- space. 2 CUBO 10, 1 (2008) On Some What Fuzzy Faintly Semicontinuous Functions 41 Proposition 4.8 Let X1, X2, Y1 and Y2 be fuzzy topological spaces such that Y1 is a product related to Y2 and X1 is a product related to X2. Then the product f1 ×f2 : X1 ×X2 → Y1 ×Y2 of some what fuzzy faintly semicontinuous mapping f1 : X1 → Y1 and f2 : X2 → Y2 is some what fuzzy faintly semicontinuous. Proof. Let α = ∨ (αi × αj ) , where αi’s and αj ’s are fuzzy θ-open sets of Y1 and Y2 respec- tively be a fuzzy θ-open set of Y1 × Y2. Then we have (f1 × f2) −1 (a) = ∨ ( f −1 1 (αi) × f −1 2 (αj ) ) Since f1 and f2 are somewhat fuzzy faintly semicontinuous functions, there exists fuzzy semiopen sets σi in X1, fuzzy semiopen set σj in X2 such that σi ≤ f −1 1 (αi) ; σj ≤ f −1 2 (αj ) . Therefore (f1 × f2) −1 (α) ≥ ∨ (σi × σj ) . Since X1 and X2 are product related, it follows from [3] that ∨ (σi × σj ) is fuzzy semiopen. Hence f1 × f2 is some what fuzzy faintly semicontinuous. 2 Received: December 2005. Revised: October 2007. References [1] A. Mukherjee, Fuzzy faintly continuous functions, Fuzzy Sets and Systems, 59 (1993), 59–63. [2] A. Mukherjee and S. Debnath, On fuzzy faintly semicontinuous functions, Indian J. pure appl. Math., 34(2003), 1625–1630. [3] A.S. Bin Shahna, On fuzzy strong semicontinuity and fuzzy precontinunity, Fuzzy Sets and Systems, 44 (1991), 303–308. [4] C.L. Chang, Fuzzy topological spaces, J.Math. Anal.Appl., 141 (1968), 82–89. [5] K.R. Gentry and H.B. Holye, Some what continuous functions, Czech. Math. J., 21(1971), 5–12. [6] G. Thangaraj and G. Balasubramanian, On some what fuzzy continuous functions, J. Fuzzy Math., 11 (2003), 1-12. [7] G. Thangaraj and G. Balasubramanian, On some what fuzzy semicontinuous func- tions, Kybernetika, 37 (2001), 165–170. [8] L.A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338–353. semi.pdf