KohPraAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 7, No. 2, 2006 pp. 177-189 Fuzzy Uniformities on Function Spaces J. K. Kohli and A. R. Prasannan ∗ Abstract. We study several uniformities on a function space and show that the fuzzy topology associated with the fuzzy unifor- mity of uniform convergence is jointly fuzzy continuous on Cf (X, Y ) , the collection of all fuzzy continuous functions from a fuzzy topologi- cal space X into a fuzzy uniform space Y . We define fuzzy uniformity of uniform convergence on starplus-compacta and show that its corre- sponding fuzzy topology is the starplus-compact open fuzzy topology. Moreover, we introduce the notion of fuzzy equicontinuity and fuzzy uniform equicontinuity on fuzzy subsets of a function space and study their properties. 2000 AMS Classification: 03E72, 04A72, 54A40, 54C35, 54D30, 54E15. Keywords: Starplus-compact open fuzzy topology, fuzzy uniformity of uni- form convergence, jointly fuzzy continuous fuzzy topology, fuzzy uniformity of uniform convergence on starplus-compacta, fuzzy equicontinuity, fuzzy uniform equicontinuity. 1. Introduction The notion of a uniform space was introduced by Andre Weil [18] in 1937. The first systematic exposition of the theory of uniform spaces was given by Bourbaki [4] in 1940. Weil elaborated the topology associated with a uni- formity and proved that a topological space is uniformizable if and only if it is completely regular. He extended the notion of uniform continuity and uniform isomorphisms to the framework of uniform spaces and obtained the uniform space version of Alexandroff-Uryshon metrization theorem that a uni- form space is metrizable if and only if its uniformity has a countable base. The concept of function space was evolved around the close of the nineteenth cen- tury and the study of function spaces began with the work of Ascoli [3], Arzelà [2] and Hadamard [8]. The uniformity of pointwise convergence and uniform ∗Corresponding author. 178 J. K. Kohli and A. R. Prasannan convergence were first defined and studied by Fox [7]. The term function space is introduced much earlier in connection with questions of a topological na- ture about sets of functions. The study of topologies on function spaces is an active area of research and, besides their multifaceted applications, forms a well-established and sound body of knowledge. The study of useful fuzzy topologies and uniformities on function spaces, be- sides their intrinsic interest, is important from applications viewpoint. The first effort in this direction was made by Peng [17] in 1984. Subsequently, Alderton [1] studied the problem from categorical viewpoint and utilized the well-developed theory of cartesian closedness of monotopological categories to the fuzzy topologies on a function space. Burton [6] defined analogues of the uniformities of pointwise convergence and uniform convergence and obtained conditions for completeness and compactness of fuzzy subsets of a function space. Jäger considered fuzzy uniform convergence and equicontinuity in [9]. In [13], we defined three different fuzzy topologies on a function space, which are analogues of the topology of pointwise convergence, compact-open topology and the topology of joint continuity in general topology. In this paper, we elaborate on the pointwise fuzzy uniformity, fuzzy unifor- mity of uniform convergence and fuzzy uniformity of uniform convergence on starplus compacta and their associated fuzzy topologies on a function space. It turns out that the fuzzy topology associated with the fuzzy uniformity of uni- form convergence is jointly continuous; and that the fuzzy topology of uniform convergence on starplus compacta is the starplus-compact open fuzzy topology [13]. Further, we study the notion of fuzzy equicontinuity on fuzzy subsets of a family of functions from a fts/fuzzy uniform space to a fuzzy uniform space. 2. Preliminaries Throughout the paper the closed unit interval [0, 1] will be denoted by I. The symbols and I0 and I1 will stand for the intervals (0, 1] and [0, 1), respectively. Definition 2.1. For a fuzzyy set µ in X, the set µα = {x ∈ X : µ(x) > α} and µα = {x ∈ X : µ(x) ≥ α} are called the strong α -level set of µ and the weak α-level set of µ , respectively. The set {x ∈ X : µ(x) > 0} is called the support of µ and is denoted by suppµ . Definition 2.2 ([13]). A fuzzy set µ in a fts (X, τ ) is said to be starplus- compact if µα is compact in (X, iα(τ )) for each α ∈ I1. The fts (X, τ ) is said to be starplus-compact if (X, iα(τ )) is compact for each α ∈ I1. Let X be a non-empty set and let (X, τ ) be a fuzzy topological space. Let Y X denote the collection of all functions from X into Y and let ℑ be a nonempty subset of Y X . Definition 2.3 ([12]). For each x ∈ X, let the map ex : ℑ −→ Y be defined by ex(f ) = f (x). We call ex the evaluation map at x ∈ X . The initial fuzzy topology on ℑ generated by the collection of maps {ex : x ∈ X} is called the Fuzzy Function spaces 179 pointwise fuzzy topology on ℑ and is denoted by τp. The pair (ℑ, τp) is called the pointwise fuzzy function space. The pointwise fuzzy topology on ℑ concides with the subspace fuzzy topolology it inherits as a subspace of the product fuzzy topology on Y X . . Definition 2.4 ([13]). Let (X, τ ) and (Y, σ) be fts and let ℑ be a nonempty subset of Y X . For each starplus-compact fuzzy set κ in X and each open fuzzy set µ in Y , define a fuzzy set κµ on ℑ by κµ = ∧ x∈suppκ e−1x (µ). The collection of all fuzzy sets κµ, where κ is a starplus-compact fuzzy set in X and µ is an open fuzzy set in Y , forms a subbase for a fuzzy topology τ + ∗C on ℑ called the starplus-compact open fuzzy topology on ℑ. The pair (ℑ, τ + ∗C ) is referred to as a starplus-compact open fuzzy function space. Proposition 2.5 ([13]). The starplus-compact open fuzzy topology on ℑ is stronger than the pointwise fuzzy topology τp on ℑ. Theorem 2.6 ([13]). Let (X, τX ) be a topologically generated fts and let (Y, τY ) be a fts. Then a fuzzy topology τ is a starplus-compact open fuzzy topology on ℑ if and only if iα(τ ) = T α C for each α ∈ I1, where T α C denotes the compact open topology on ℑ and X is endowed with the topology i0(τX ) and Y is equipped with the topology iα(τY ). Definition 2.7 ([13]). A fuzzy topology τ on ℑ such that the map φ : ℑ×X −→ Y defined by φ(f, x) = f (x) is fuzzy continuous, where ℑ × X is endowed with the product fuzzy topology, is called a jointly fuzzy continuous fuzzy topology on ℑ. Theorem 2.8 ([13]). The fuzzy topology of joint fuzzy continuity is a good extension. Definition 2.9 ([15]). A subset F ⊂ IX is called a prefilter if and only if F 6= φ, and i) For all µ, ν ∈ F we have µ ∧ ν ∈ F. ii) If µ ≥ ν and ν ∈ F, then µ ∈ F. iii) 0 6∈ F. Definition 2.10 ([15]). A subset B ⊂ IX is a base for a prefilter if and only if B 6= φ, and i) For all µ, ν ∈ B there exists a ξ ∈ B such that ξ ≤ µ ∧ ν. ii) 0 6∈ B. Definition 2.11 ([15]). A prefilter generated by a prefilter base B is denoted as 〈B〉 and 〈B〉 = {µ ∈ IX : there exists a ν ∈ B such that µ ≥ ν}. If B is a prefilter base, then B̂ = {sup ǫ∈I0 (βǫ − ǫ) : (βǫ)ǫ∈I0 ∈ B I0}. Proposition 2.12 ([15]). If B is a prefilter base, then 〈B̂〉= 〈B〉. We shall denote by B̃ the prefilter 〈B̂〉 = 〈B〉. 180 J. K. Kohli and A. R. Prasannan Definition 2.13 ([15]). A prefilter F is called prime if µ ∨ ν ∈ F implies µ ∈ F or ν ∈ F. Definition 2.14 ([15]). If F is a prefilter on X, then we define the following: P (F) = {G : G is a prime prefilter and F ⊂ G} and Pm(F) = {G : G ∈ P (F) and G is minimal}. Definition 2.15 ([15]). For a prefilter F the characteristic of F is defined by c(F) = inf ν∈F sup ν. For a prefilter F the lower characteristic of F is defined by c(F) = inf G∈Pm(F) c(G). If F is a prime, then c(F) = c(F). Definition 2.16 ([16]). If X is a set, µ ∈ IX and ν ∈ IX×X , then the section of ν over µ is defined by ν〈µ〉(x) = sup y∈X µ(y) ∧ ν(y, x) for all x ∈ X. Definition 2.17 ([16]). If µ, ν ∈ IX×X , then the composition µ◦ν is defined by µ ◦ ν(x, y) = sup z∈X ν(x, z) ∧ µ(z, y) for all (x, y) ∈ X × X. Definition 2.18 ([16]). If ν ∈ IX×X , then its symmetric sν ∈ I X×X is defined by sν(x, y) = ν(y, x) for all (x, y) ∈ X × X. Throughout this paper we follow the terminology and notions of a fuzzy uni- formity as defined by Lowen [16]. Definition 2.19 ([16]). A fuzzy uniformity on X is a subset U ⊂ IX×X , which satisfies the following conditions: i) U is a prefilter. ii) Û = U, i.e., for every family (νǫ)ǫ∈I0 ∈ U I0 =⇒ sup ǫ∈I0 (νǫ − ǫ) ∈ U. iii) For all ν ∈ U and for all x ∈ X, ν(x, x) = 1. iv) For all ν ∈ U, sν ∈ U. v) For all ν ∈ U and for all ǫ ∈ I0 there exists νǫ ∈ U such that νǫ◦νǫ−ǫ ≤ ν. The pair (X, U) is called a fuzzy uniform space. Definition 2.20 ([16]). A subset B ⊂ IX×X is called a base for a fuzzy uniformity if and only if the following conditions hold: i) B is a prefilter basis. ii) For all β ∈ B and for all x ∈ X, β(x, x) = 1. iii) For all β ∈ B and for all ǫ ∈ I0, there exists βǫ ∈ B such that βǫ − ǫ ≤ sβ. iv) For all β ∈ B and for all ǫ ∈ I0, there exists βǫ ∈ B such that βǫ ◦ βǫ − ǫ ≤ β. Definition 2.21. If U is a fuzzy uniformity on X then B ⊂ IX×X is a basis for U iff B is a prefilter basis and B̃ = U. Proposition 2.22. If U is a fuzzy uniformity on X, then the family of sym- metric fuzzy entourages sU = {ν ∈ U : sν = ν} is a basis for U. Fuzzy Function spaces 181 Definition 2.23. Let (X, τ ) be a fts. Then the closure µ of a fuzzy set µ of X is defined as µ = inf{ν : µ ≤ ν, 1 − ν ∈ τ}. Definition 2.24 ([14]). A fuzzy closure operator on a fts X is a map ¯: IX −→ IX which satisfies the following conditions: i) α = α, for all α ∈ I. ii) µ ≥ µ, for all µ ∈ IX . iii) µ ∨ ν = µ ∨ ν, for all µ, ν ∈ IX . iv) µ = µ, for all µ ∈ IX . Proposition 2.25 ([16]). Let (X, U ) be a fuzzy uniform space. The map¯: IX −→ IX defined by µ= inf ν∈U ν〈µ〉 is a fuzzy closure operator. Definition 2.26 ([5]). If F is a prefilter on (X, U), then AdhF and lim F are fuzzy sets in X and is defined by AdhF = inf ν∈F ν and lim F = inf G∈Pm(F) AdhG. If µ ∈ IX , we say that F is U-convergent in µ iff c(F) 6 sup µ ∧ lim F and F is U-convergent iff c(F) 6 sup lim F . Definition 2.27. Let (X, U) be a fuzzy uniform space and F is a prefilter on X. Then F is U-cauchy iff c(F) 6 inf σ∈U sup inf G∈Pm(F) inf ν∈G σ〈ν〉. Definition 2.28. Let (X, U) and (Y, U1) be fuzzy uniform spaces. Then a map f : X −→ Y is said to be fuzzy uniformly continuous if for each ν ∈ U1, (f × f )−1(ν) ∈ U. Proposition 2.29. If (X, U) and (Y, U1) are fuzzy uniform spaces, B and B1 are basis for U and U1, respectively and f : X −→ Y , then f is fuzzy uniformly continuous iff for all β1 ∈ B1 and for all ε ∈ I0 there exist β ∈ B such that β − ε ≤ (f × f )−1(β1). Theorem 2.30. If (X, U) and (Y, U1) are fuzzy uniform space and f : X −→ Y is uniformly continuous, then f is fuzzy continuous. Throughout this paper uniformity on a nonempty set X is denoted by U and a fuzzy uniformity by U. We denote the topology associated with a uniformity U by T (U) and the fuzzy topology associated with a fuzzy uniformity U by τ (U) [16], where τ (U) is the fuzzy topology whose fuzzy closure operator is defined in Proposition 2.25. Definition 2.31 ([16]). Let X be a non empty set and let {fj : X → (Yj , Uj ), j ∈ J} be a family of functions from X into the family of fuzzy uniform spaces {(Yj , Uj), j ∈ J}. Then the coarsest fuzzy uniformity U on X making each fj, j ∈ J is fuzzy uniformly continuous is called the initial fuzzy uniformity on X and is denoted by sup j∈J (fj × fj) −1(Uj ). Let UNIF denote the category of uniform spaces and uniformly continuous functions and FUNIF denote the category of fuzzy uniform spaces and fuzzy uniformly continuous functions. Then the functors ωu : UNIF → FUNIF and 182 J. K. Kohli and A. R. Prasannan iu : FUNIF → UNIF are defined as follows and are the uniform analogues of the functors ω and i introduced in [14]. For each (X, U) ∈ UNIF, ωu(X, U) = (X, ωu(U)), where ωu(U) = {µ ∈ IX×X : µ−1(α, 1] ∈ U, ∀α ∈ I1} and for each (X, U) ∈ FUNIF, iu(X, U) = (X, iu(U)), where iu(U) = {µ −1(α, 1] : µ ∈ U, α ∈ I1} . Theorem 2.32 ([16]). If U is a uniformity on X and U a fuzzy uniformity on X. Then, 1) ωu(U) is a fuzzy uniformity on X. 2) iu(U) is a uniformity on X. 3) iu(ωu(U)) = U. 4) ωu(iu(U)) is the coarsest fuzzy uniformity generated by a uniformity and is finer than U. We denote ωu(iu(U)) by U. 5) τ (ωu(U))= ω(T (U)). 6) T (iu(U)) = i(τ (U). 7) τ (U) = τ (U). We shall call a notion in FUNIF a good extension of a notion in UNIF if it reduces to the standard notion in case of the fuzzy uniformity U = ω(U). Definition 2.33 ([10]). If U is a fuzzy uniformity on a nonempty set X, then its α-level uniformity, for 0 ≤ α ≤ 1 is defined by iu,α(U) = {µ β ∈ 2X×X : µ ∈ U, β ∈ [0, 1 − α)} . The functor iu,α : FUNIF −→ UNIF is the uniform analogue of the functor iα discussed in [16]. Also, iu = iu,0 . Theorem 2.34 ([10]). The topology T (iu,α(U)) on X, induced by the α-level uniformity iu,α(U) of the fuzzy uniform space (X, U), coincides with the α-level topology iα(τ (U)). 3. Fuzzy Uniformities on Function Spaces Let X be a non-empty set and let (Y, U) be a fuzzy uniform space. Let Y X denote the collection of maps from X into Y . Let ℑ be a nonempty subset of Y X . In this section we study the pointwise fuzzy uniformity and fuzzy uniformity of uniform convergence and their associated fuzzy topologies. It is shown that the fuzzy topology associated with the fuzzy uniformity of uniform convergence is jointly fuzzy continuous on Cf (X, Y ). Definition 3.1. The initial fuzzy uniformity Up on ℑ generated by the col- lection of maps {ex : x ∈ X} is called the pointwise fuzzy uniformity or fuzzy uniformity of pointwise convergence on ℑ. The pair (ℑ, Up) is called the pointwise fuzzy uniform space. Remark 3.2. The above definition of pointwise fuzzy uniformity on ℑ coin- cides with the definitions of pointwise fuzzy uniformity given in [6, 9]. Fuzzy Function spaces 183 The following theorem of Lowen [16] reflects upon the relationship that exists between the initial fuzzy topology and the fuzzy topology generated by the initial fuzzy uniformity. Theorem 3.3 ([16]). Let X be a set and let (Y, Uj ), j ∈ J be a family of fuzzy uniform spaces. If {fj : X −→ Yj , j ∈ J} is a family of functions, then τ (sup j∈J (fj × fj ) −1(Uj )) = sup j∈J f −1 j (τ (Uj )). In view of Definitions 2.3, 3.1 and Theorem 3.3, the following result is imme- diate. Theorem 3.4. The fuzzy topology associated with the fuzzy uniformity Up of pointwise fuzzy uniformity is the pointwise fuzzy topology τ P . Theorem 3.5. Let X be a non-empty set and let (Y, U) be a uniform space. Let Up denote the pointwise uniformity on ℑ and let Up be the pointwise fuzzy uniformity on ℑ, where Y is endowed with the fuzzy uniformity ωu(U). Then ωu(Up) = Up. Proof. Let n∧ i=1 (ex × ex) −1(νi) be a basic element for the fuzzy uniformity Up, where Y is endowed with the fuzzy uniformity ωu(U). Since νi ∈ ωu(U), ν α i ∈ U for each α ∈ I1 and hence [ n⋂ i=1 (ex × ex) −1(ναi )] ∈ Up. This shows that [ n∧ i=1 (ex × ex) −1(νi] α ∈ Up and so n∧ i=1 (ex × ex) −1(νi) ∈ ωu(Up). Thus we have ωu(Up) ⊇ Up. The proof of the opposite inclusion, ωu(Up) ⊆ Up, is similar to the one given above. � Proposition 3.6. Let X be a set and let (Y, U) be a fuzzy uniform space. Let Up be the fuzzy uniformity of pointwise convergence on ℑ . Then for each α ∈ I1, the α-level uniformity iu,α(Up) is the uniformity of pointwise convergence on ℑ with respect to iu,α(U) on Y . Proof. Let (ex × ex) −1(ν) be a subbasic element in Up. Then for each α ∈ I1, [(ex × ex) −1(ν)]β = {(f, g) ∈ ℑ × ℑ : ν(f (x), g(x)) > β}, β ∈ [0, 1 − α) = {(f, g) ∈ ℑ × ℑ : (f (x), g(x)) ∈ νβ} = (ex × ex) −1(νβ ), which is a subbasic element in the pointwise uniformity on ℑ, where Y is endowed with the uniformity iu,α(U). � Theorem 3.7 ([6]). Let F be a prefilter in the pointwise fuzzy uniform space (Y X , Up). Then F is a Cauchy prefilter if and only if for each x ∈ X, ex(F) is a Cauchy prefilter in (Y, UY ). 184 J. K. Kohli and A. R. Prasannan Fuzzy Uniformity of Uniform Convergence. In this subsection we study the notion of fuzzy uniformity of uniform conver- gence on a function space ℑ ⊂ Y X (which was initiated by Burton [6]), where X is a nonempty set and (Y, U) is a fuzzy uniform space. Definition 3.8 ([6]). For each ν ∈ U, the fuzzy set Wν = ∧ x∈X (ex × ex) −1(ν) in ℑ × ℑ, where ex : ℑ −→ Y is the evaluation map, is defined by, Wν (f, g) = ∧ x∈X ν(f (x), g(x)). Let B be the collection of all Wν , where ν varies over U. Proposition 3.9 ([6]). The collection B is a base for a fuzzy uniformity on ℑ. Example 3.10. If µ ∈ Iℑ, then the section of Wν over µ is defined by Wν〈µ〉(f ) = ∧ x∈X ν〈ex(µ)〉(f (x)) = inf x∈X {sup g∈ℑ (ex(µ)(g(x)) ∧ ν(g(x), f (x)))} for each f ∈ ℑ. Definition 3.11 ([6]). The fuzzy uniformity Uu generated by B is called the fuzzy uniformity of uniform convergence. The fuzzy topology associated with Uu is called the fuzzy topology of uniform convergence and it is denoted by τu. In the following results we give a short description of the concepts that Burton [6] uses relative to the fuzzy uniform convergence. Theorem 3.12 ([6]). Let F be a prefilter in Y X with c(F) = c(F) . Then F is Uu-cauchy, α ≤ c, lim Up (F)(f ) ≥ α =⇒ lim Uu (F)(f ) ≥ α . Theorem 3.13 ([6]). If (X, U) is complete then (Y X , Uu) is complete. Corollary 3.14 ([6]). If µ : Y X −→ I is Up-closed and (Y, U) is complete, then µ is Uu-complete. Theorem 3.15 ([6]). If µ is a closed fuzzy set in (Y X , Up) and for all x ∈ X, ex(µ) is complete in (X, U) , then the fuzzy set µ is complete in (Y X , Uu). Proposition 3.16. If Wνi , 1 ≤ i ≤ n are members of Uu. Then the following hold: i) n∧ i=1 Wνi = W ( n V i=1 νi) and ii) W α ν = Wνα , for α ∈ I1. Fuzzy Function spaces 185 Proof. i) For each i, Wνi = ∧ x∈X (ex × ex) −1(νi). Hence, n∧ i=1 Wνi = n∧ i=1 { ∧ x∈X (ex × ex) −1(νi)} = ∧ x∈X [(ex × ex) −1( n∧ i=1 νi) = W ( n V i=1 νi) . ii) For each α ∈ I1, Wνα = ⋂ x∈X {(f, g) : (f (x), g(x)) ∈ να} = ⋂ x∈X {(f, g) : (ex × ex)(f, g) ∈ ν α} = ⋂ x∈X {(f, g) : (f, g) ∈ (ex × ex) −1(να)} = ⋂ x∈X {(f, g) : (f, g) ∈ [(ex × ex) −1(ν)]α} = ⋂ x∈X [(ex × ex) −1(ν)]α = [ ∧ x∈X (ex × ex) −1(ν)]α = (Wν ) α. � The following theorem shows that for each α ∈ I1, the α-level uniformity of the fuzzy uniformity of uniform convergence Uu on the function space ℑ coincides with the uniformity of uniform convergence on ℑ when Y is endowed with the uniformity iu,α(U). Theorem 3.17. For each α ∈ I1, iu,α(Uu) is the uniformity of uniform con- vergence on ℑ, where Y is endowed with the uniformity iu,α(U). Proof. Let Wν be a basic element for the fuzzy uniformity of uniform con- vergence Uu, where ν ∈ U. Then W α ν is a basic element for the uniformity iu,α(Uu). By Proposition 3.16(ii), Wνα = W α ν and the fact that Wνα is a basic element for the uniformity of uniform convergence on ℑ, where Y is endowed with the uniformity iu,α(Uu), the theorem follows. � Jäger in [9] showed that the notion of fuzzy uniformity of uniform convergence is a good extension. 186 J. K. Kohli and A. R. Prasannan Theorem 3.18. Let C(X, Y ) denote the collection of all continuous functions from a topological space X into a uniform space Y and let Uu denote the unifor- mity of uniform convergence on C(X, Y ). Then the fuzzy topology associated with the fuzzy uniformity ωu(Uu) is jointly fuzzy continuous. Proof. The uniform topology T (Uu) on C(X, Y ) is jointly continuous. Since in view of Theorem 2.8 the fuzzy topology of joint fuzzy continuity is a good ex- tension, the fuzzy topology ω(T (Uu)) is jointly fuzzy continuous. By Theorem 2.32, τ (ωu(Uu)) = ω(T (Uu)). This shows that the fuzzy topology τ (ωu(Uu)) associated with the fuzzy uniformity ωu(Uu) is jointly fuzzy continuous. � Corollary 3.19. Let X be a topological space and let Y be a uniform space with the uniformity UY . Let Cf (X, Y ) denote the collection of all fuzzy continuous maps from the topologically generated fts ω(X) into the topologically generated fts ω(Y ). Then the fuzzy topology of uniform convergence on Cf (X, Y ) is jointly fuzzy continuous. Proof. Since X and Y are topologically generated fts, Cf (X, Y ) and C(X, Y ) are equal as sets. Since the uniformity of uniform convergence is a good ex- tension, the fuzzy uniformity of uniform convergence Uu is same as ωu(Uu), where Uu is the uniformity of uniform convergence on C(X, Y ). Hence in view of Theorem 3.18, the fuzzy topology associated with the fuzzy uniformity of uniform convergence Uu is jointly fuzzy continuous. � Definition 3.20. Let X be a fuzzy topological space and let (Y, U) be a fuzzy uniform space. Let S be the collection of all starplus-compact fuzzy sets in X. For each κ ∈ S and ν ∈ U, define a fuzzy set W(κ, ν) : ℑ × ℑ −→ I by W(κ, ν)(f, g) = ∧ x∈suppκ (ex × ex) −1(ν)(f, g) = ∧ x∈suppκ ν(f (x), g(x)). Then the collection of all fuzzy sets {W(κ, ν) : κ ∈ S, ν ∈ U} is a base for a fuzzy uniformity on ℑ and is called the fuzzy uniformity of uniform convergence on starplus-compacta. Theorem 3.21. Let ℑ be the set of all fuzzy continuous maps from a topolog- ically generated fts (X, τX ) into a fuzzy uniform space (Y, U). Then the fuzzy topology of fuzzy uniform convergence on starplus-compacta is the starplus- compact open fuzzy topology. Proof. Let U+ ∗C be the fuzzy uniformity of uniform convergence on starplus- compacta. Since X is a topologically generated fts, a subset K of X is com- pact in (X, i0(τX )) if and only if χK is starplus-compact in (X, τX ). Hence iuα(U + ∗C ) = UUC , where UUC denotes the uniformity of uniform convergence on compacta, where X is endowed with the topology i0(τX ) and Y is equipped with the uniformity iuα(U). So by [11, Theorem 7.11], T (iuα(U + ∗C )) is the com- pact open topology for each α ∈ I1. By Theorem 2.6, T (iuα(U + ∗C )) = iα(τ + ∗C ) Fuzzy Function spaces 187 for each α ∈ I1. Again by Theorem 2.34, iα(τ (U + ∗C )) = T (iuα(U + ∗C )) = iα(τ + ∗C ) for each α ∈ I1. This completes the proof. � 4. Fuzzy Equicontinuity In this section we introduce the notion of fuzzy equicontinuity and fuzzy uni- form equicontinuity on fuzzy subsets of ℑ and obtain results, which show that if a fuzzy subset κ of Y X is fuzzy equicontinuous (respectively, fuzzy uniformly equicontinuous), then each f ∈ suppκ is fuzzy continuous (respectively, fuzzy uniformly continuous). Definition 4.1. Let (X, τ ) be a fts and let (Y, U) be a fuzzy uniform space. Then a fuzzy subset κ of Y X is said to be fuzzy equicontinuous at a fuzzy point xα of X if for each µ ∈ U, there is a τ -neighbourhood η of xα such that f (η) ≤ µ〈f (xα)〉 for each f ∈ suppκ. We say that the fuzzy subset κ of Y X is fuzzy equicontinuous on a fuzzy subset θ of X if κ is fuzzy equicontinuous at each fuzzy point xα in θ. Definition 4.2. A fuzzy subset κ of Y X , where (X, UX ) and (Y, UY ) are fuzzy uniform spaces, is said to be fuzzy uniformly equicontinuous if ∧ f∈suppκ (f × f )−1(µ) ∈ UX for each µ ∈ UY . The following is a characterization of fuzzy equicontinuity. Theorem 4.3. A fuzzy subset κ of Y X is fuzzy equicontinuous if and only if for each µ ∈ UY , the fuzzy set ∧ f∈suppκ f −1(µ〈f (xα)〉) is a neighbourhood of xα. Proof. Suppose that a fuzzy subset κ of Y X is fuzzy equicontinuous at a fuzzy point xα of X. Then for each µ ∈ UY , there is a τ -neighbourhood η of xα such that f (η) ≤ µ〈f (xα)〉 for each f ∈ suppκ. So η ≤ ∧ f∈suppκ f −1(µ〈f (xα)〉) and hence ∧ f∈suppκ f −1(µ〈f (xα)〉) is a neighbourhood of xα. Conversely, suppose that ∧ f∈suppκ f −1(µ〈f (xα)〉) is a neighbourhood of the fuzzy point xα for each µ ∈ UY . Let η = ∧ f∈suppκ f −1(µ〈f (xα)〉). Then clearly g(η) ≤ µ〈g(xα)〉 for each g ∈ suppκ and so κ is fuzzy equicontinuous at the fuzzy point xα. � Theorem 4.4. If a fuzzy set κ in Y X is fuzzy equicontinuous, then suppκ is equicontinuous, where X is endowed with the topology i0(τX ) and Y is equipped with the uniformity iu,α(UY ). Proof. Let κ be fuzzy equicontinuous. Then for each fuzzy point xλ in X and for each µ ∈ UY , there exist a neighbourhood η of xλ in X such that f (η) ≤ µ〈f (xλ)〉 for each f ∈ suppκ. Hence in particular, f (η) ≤ µ〈f (x1)〉 with λ = 1. This shows that f (ηβ ) ⊂ µβ〈f (x)〉, β ∈ [0, 1 − α). Thus for each 188 J. K. Kohli and A. R. Prasannan x ∈ X and µβ ∈ iu,α(UY ), there exist a neighbourhood η β of x such that f (ηβ) ⊂ µβ〈f (x)〉, for each f ∈ suppκ. Hence suppκ is equicontinuous. � Theorem 4.5. If a fuzzy subset κ in Y X is fuzzy equicontinuous, then κα is equicontinuous for each α ∈ I1, where Y is equipped with the uniformity iu,α(UY ). First we prove the following lemma. Lemma 4.6. If ν ≤ κ and κ is fuzzy equicontinuous then ν is also fuzzy equicontinuous. Proof. Since ν ≤ κ, suppν ⊂ suppκ. Hence ∧ f∈suppκ f −1(µ〈f (xα)〉) ≤ ∧ f∈suppν f −1(µ〈f (xα)〉) and so the result fol- lows. � Proof of Theorem 4.5. Since κα ⊂ suppκ for each α ∈ I1 and since κ is fuzzy equicontinuous then by Theorem 4.4 and Lemma 4.6, κα is equicontinuous.2. Theorem 4.7. If a fuzzy subset κ of Y X is fuzzy equicontinuous, then each f ∈ suppκ is fuzzy continuous. Proof. Since ∧ f∈suppκ f −1(µ〈f (xα)〉) ≤ f −1(µ〈f (xα)〉) for each fuzzy point xα of X, f −1(µ〈f (xα)〉) is a neighbourhood of xα for each µ ∈ UY and so f is fuzzy continuous at each xα. Hence f is fuzzy continuous on X. � Theorem 4.8. If κ is fuzzy uniformly equicontinuous, then each f ∈ suppκ is fuzzy uniformly continuous. Proof. Suppose that κ is fuzzy uniformly equicontinuous. Then for each µ ∈ UY , ∧ f∈suppκ (f × f )−1(µ) ∈ UX . This implies that for each µ ∈ UY , (f × f )−1(µ) ∈ UX . Hence f is fuzzy uniformly continuous. � Acknowledgements. The authors are thankful to the referee for helpful suggestions. References [1] I. W. Alderton, Function spaces in fuzzy topology, Fuzzy Sets and Systems 32 (1989), 115-124. [2] C. Arzelà, Funzioni di Linee, Atti della Reale Accademia dei Lincei, Rendiconti 5 (1889), 342–348. [3] G. Ascoli, Le Curve Limite di una Varieta Data di Curve, Mem. Acad. Lincei (3) 18(1883), 512–586. Fuzzy Function spaces 189 [4] N. 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Prasannan, Starplus-compactness in fuzzy topological spaces and starplus-compact open fuzzy topologies on function spaces, J. Math. Anal. Appl. 254 (2001), 87–100. [14] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976), 621–633. [15] R. Lowen, Convergence in fuzzy topological spaces, Gen. Top. Appl. 10 (1979), 147–160. [16] R. Lowen, Fuzzy uniform spaces, J. Math. Anal. Appl. 82 (1981), 370–385. [17] Y. W. Peng, Topological structure of a fuzzy function space- the pointwise convergent topology and compact open topology, Kexue Tongbao (English Ed.) 29 (1984), 289–292. [18] A.Weil, Sur les Espaces à Structure Uniform et sur la Topologie Générale, Act.Sci.et Ind. 551, Hermann, Paris (1937). Received April 2005 Accepted March 2006 J. K. Kohli Department of Mathematics, Hindu College, University of Delhi, Delhi - 110 007, India. A. R. Prasannan (arprasannan@yahoo.co.in) Department of Mathematics, Maharaja Agrasen College, University of Delhi, Pocket - IV, Mayur Vihar - I, Delhi - 110 091, India.