@ Appl. Gen. Topol. 15, no. 2(2014), 155-166doi:10.4995/agt.2014.3029 c© AGT, UPV, 2014 Rcl-spaces and closedness/completeness of certain function spaces in the topology of uniform convergence J. K. Kohli a and D. Singh b a Department of Mathematics, Hindu College, University of Delhi, Delhi, India (jk kohli@yahoo.com) b Department of Mathematics, Sri Aurobindo College, University of Delhi, Delhi, India. (dstopology@rediffmail.com) Abstract It is shown that the notion of an Rcl-space (Demonstratio Math. 46(1) (2013), 229-244) fits well as a separation axiom between zero dimen- sionality and R0-spaces. Basic properties of Rcl-spaces are studied and their place in the hierarchy of separation axioms that already exist in the literature is elaborated. The category of Rcl-spaces and continuous maps constitutes a full isomorphism closed, monoreflective (epireflective) subcategory of TOP. The function space Rcl(X, Y) of all Rcl-supercontinuous functions from a space X into a uniform space Y is shown to be closed in the topology of uniform convergence. This strengthens and extends certain results in the literature (Demonstratio Math. 45(4) (2012), 947-952). 2010 MSC: 54C08; 54C10; 54C35; 54D05; 54D10. Keywords: Rcl-space; ultra Hausdorff space; initial property; monoreflec- tive (epireflective) subcategory; Rcl-supercontinuous function; topology of uniform convergence. 1. introduction The notion of an Rcl-space evolved naturally in the study of Rcl-supercontinuous functions [37]. Here we study their basic properties and show that it fits well as a separation axiom between zero dimensionality and R0-spaces. We reflect Received 13 July 2013 – Accepted 20 May 2014 http://dx.doi.org/10.4995/agt.2014.3029 J. K. Kohli and D. Singh upon interrelations and interconnections that exist among Rcl-spaces and sep- aration axioms which already exist in the lore of mathematical literature and lie between zero dimensionality and R0-spaces. The class of Rcl-spaces prop- erly contains each of the classes of zero dimensional spaces and ultra Hausdorff spaces [35] and is strictly contained in the class of R0-spaces ([20, 33]) which in its turn properly contains each of the classes of functionally regular spaces ([3, 39]) and functionally Hausdorff spaces. The organization of the paper is as follows: Section 2 is devoted to preliminaries and basic definitions. In Section 3 we elaborate upon the place of Rcl-spaces in the hierarchy of separation axioms which lie between zero dimensionality and R0-spaces and already exist in the mathematical literature. Section 4 is devoted to study basic properties of Rcl-spaces wherein it is shown that (i) the property of being an Rcl-spaces is invariant under disjoint topological sums and initial sources so it is hereditary, productive, supinvariant, preimage invariant and projective; (ii) the category of Rcl-spaces and continuous maps is a full, isomorphism closed monoreflective (epireflective) subcategory of TOP; (iii) it is shown that a T0-space is ultra Hausdorff if and only if it is an Rcl-space. In Section 5 we discuss the relation between Rcl-supercontinuous functions and Rcl-spaces. Section 6 is devoted to the study of function spaces wherein it is shown that the function space of all Rcl(X, Y ) of all Rcl-supercontinuous func- tions from a topological space X into a uniform space Y is closed in Y X in the topology of uniform convergence and the condition for its completeness is outlined. 2. Preliminaries and basic definitions Let X be a topological space. A subset A of a space X is called regular Gδ-set [23] if A is an intersection of a sequence of closed sets whose interiors contain A, i.e., if A = ∞⋂ n=1 Fn = ∞⋂ n=1 F 0 n ,where each Fn is a closed subset of X (here F 0 n denotes the interior of Fn). The complement of a regular Gδ-set is called a regular Fσ-set. Any union of regular Fσ-sets is called dδ-open [17]. The complement of a dδ-open set is referred to as a dδ-closed set. A subset A of a space X is said to be regular open if it is the interior of its closure, i.e., A = A 0 . The complement of a regular open set is referred to as a regular closed set. Any union of regular open sets is called δ-open set [40]. The complement of a δ-open set is referred to as a δ-closed set. Any intersection of closed Gδ-sets is called d-closed set [16]. Any intersection of zero sets is called z-closed set ([15, 30]). A collection β of subsets of a space X is called an open complementary system [9] if β consists of open sets such that for every B ∈ β, there exist B1, B2, . . . ∈ β with B = ∪{X \ Bi : i ∈ N}. A subset A of a space X is called a strongly open Fσ-set [9] if there exists a countable open complementary system β(A) with A ∈ β(A). The complement of a strongly open Fσ-set is c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 156 Rcl-spaces and closedness/completeness of certain function spaces called strongly closed Gδ-set. Any intersection of strongly closed Gδ-sets is called d∗-closed set [31]. Definition 2.1. A topological space X is said to be (i) functionally regular ([3, 39]) if for each closed set F in X and each x /∈ F there exists a continuous real-valued function f defined on X such that f(x) /∈ f(F). (ii) ultra Hausdorff [35] if every pair of distinct points in X are contained in disjoint clopen sets. (iii) Rz-space ([20, 33]) if for each open set U in X and each x ∈ U there exists a z-closed set H containing x such that H ⊂ U; equivalently U is expressible as a union of z-closed sets. (iv) Rδ-space [19] if for each open set U in X and each x ∈ U there exists a δ-closed set H containing x such that H ⊂ U; equivalently U is expressible as a union of δ-closed sets. (v) R0-space ([5],[38] 1 [28]) if for each open set U in X and each x ∈ U implies that {x} ⊂ U. (vi) R1-space ([42] 2 [5]) if x /∈ {y} implies that x and y are contained in disjoint open sets. (vii) π2-space [38] 3 (≡ PΣ-space [41]≡ strongly s-regular space [7]) if every open set in X is expressible as a union of regular closed sets. (viii) π0-space ([38, p 98]) if every nonempty open set in X contains a nonempty closed set. Definition 2.2 ([19]). A space X is said to be an (i) RDδ -space if for each open set U in X and each x ∈ U there exists a regular Gδ-set H containing x such that H ⊂ U; equivalently U is expressible as a union of regular Gδ-sets. (ii) Rdδ -space if for each open set U in X and each x ∈ U there exists a dδ-closed set H containing x such that H ⊂ U; equivalently U is expressible as a union of dδ-closed sets. (iii) RD-space if for each open set U in X and each x ∈ U there exists a closed Gδ-set H containing x such that H ⊂ U; equivalently U is expressible as a union of closed Gδ-sets. (iv) Rd-space if for each open set U in X and each x ∈ U there exists a d-closed set H containing x such that H ⊂ U; equivalently U is expressible as a union of d-closed sets. 1Vaidyanathswamy calls R0-axiom as π1-axiom in his text book (see [38, p 98]). Császár calls an R0-space as S1-space in [4]. 2Yang [42] in his studies of paracompactness refers an R1-space as a T2-space. Császár calls an R1-space as S2-space in [4]. 3 π2-spaces were defined by Vaidyanathswamy [38] (1960) and rediscovered by Wong [41] (1981) and Ganster [7] (1990) with different terminologies. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 157 J. K. Kohli and D. Singh Definition 2.3 ([20]). A space X is said to be an (i) RD∗-space if for each open set U in X and each x ∈ U there exists a strongly closed Gδ-set H containing x such that H ⊂ U; equivalently U is expressible as a union of strongly closed Gδ-sets. (ii) Rd∗-space if for each open set U in X and each x ∈ U there exists a d∗-closed set H containing x such that H ⊂ U; equivalently U is expressible as a union of d∗-closed sets. Definition 2.4. A space X is said to be (i) D-completely regular [9] if it has a base of strongly open Fσ-sets. (ii) D-regular [9] if it has a base of open Fσ-sets. (iii) weakly regular [9] if it has a base of Fσ-neighbourhoods. (iv) Dδ-completely regular [18] if it has a base of regular Fσ-sets. 3. Rcl-spaces and hierarchy of seperation axioms Definition 3.1. Let X be a topological space. Any intersection of clopen sets in X is called cl-closed [32]. An open subset U of X is said to be rcl-open [37] if for each x ∈ U there exists a cl-closed set H containing x such that H ⊂ U; equivalently U is expressible as a union of cl-closed sets. Definition 3.2 ([37]). A topological space X is said to be an Rcl-space if every open set in X is rcl-open. It is clear from the definitions that every zero dimensional space as well as every ultra Hausdorff space is an Rcl-space. The space of strong ultrafilter topology [36, Example 113, p.133] is a Hausdorff extremally disconnected Rcl- space which is not zero dimensional. The comprehensive diagram (Figure 1) well reflects the place of Rcl-spaces in the hierarchy of separation axioms related to the theme of the present paper and certain other topological invariants and extends several existing diagrams in the literature (see [9, 18, 19]). However, most of the implications of Figure 1 are irreversible (see [9, 18, 19, 20]). We reproduce the diagram (Figure 2) from [20] concerning separation axioms between functionally regular space and R0-space, which is complemen- tary to Figure 1. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 158 Rcl-spaces and closedness/completeness of certain function spaces partition topology zero dimensional space ultra Hausdorff developable space pseudo metrizable Rcl - space perfect space perfectly normal D-completely regular completely regular functionally regular Rz - space D-regular D -completely regular dR - space regular space 2 - space ( P -space strongly s-regular space) D R - space d R - space D R - space Rd - space 1R - space weakly regular space R0 - space R -space 0 - space Figure 1. functionally regular Rz - space * D R - space D R - space d R - space D R - space 1 R - space z R - space * d R - space d R - space R - space 0 R - space 0 - space Figure 2. 4. Basic properties of Rcl-spaces Definition 4.1. Let X be a topological space. A point x ∈ X is said to be an rcl-adherent point of a set A ⊂ X if every rcl-open set containing x intersects A. Let Arcl denote the set of all rcl-adherent points of the set A. Then A ⊂ A ⊂ Arcl. The set A is rcl-closed if and only if A = Arcl. Lemma 4.2. The correspondence A → Arcl is a Kuratowski closure operator. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 159 J. K. Kohli and D. Singh Theorem 4.3. Let X be a topological space. Consider the following statements: (i) X is an Rcl-space (ii) For each x ∈ X and for each open set U containing x, {x}rcl ⊂ U (iii) There exists a subbase S for X such that x ∈ S ∈ S ⇒ {x}rcl ⊂ S (iv) x ∈ {y}rcl ⇒ y ∈ {x}rcl (v) x ∈ {y}rcl ⇒ {x}rcl = {y}rcl Then (i) ⇔ (ii) ⇔ (iii) ⇒ (iv) ⇔ (v). Proof. (i) ⇒ (ii). Let x ∈ X and let U be an open set containing x. Since X is an Rcl-space, there exists an rcl-closed set A such that x ∈ A ⊂ U. Consequently {x}rcl ⊂ U. The assertions (ii) ⇒ (i) and (ii) ⇔ (iii) are trivial. iii) ⇒ (iv). Since every subbasic open set containing y contains {y}rcl, every basic open set containing y contains {y}rcl and hence it contains x. So y ∈ {x}rcl. (iv) ⇒ (v). Since x ∈ {y}rcl, y ∈ {x}rcl. So x ∈ {y}rcl and y ∈ {x}rcl implies {x}rcl ⊂ {y}rcl and {y}rcl ⊂ {x}rcl, and hence {x}rcl = {y}rcl. The implication (v) ⇒ (iv) is obvious. � Theorem 4.4. For a topological space X the following statements are equiva- lent: (i) {x}rcl 6= {y}rcl implies that x and y are contained in disjoint open sets (ii) x /∈ {y}rcl implies that x and y are contained in disjoint open sets (iii) A is compact set and {x}rcl ∩ A = ∅ implies x and A are contained in disjoint open sets (iv) If A and B are compact sets, and {a}rcl ∩B = ∅ for every a ∈ A, then A and B are contained in disjoint open sets. Proof. (i) ⇒ (ii). Suppose that x /∈ {y}rcl. Then {x}rcl 6= {y}rcl and so by (i) x and y are contained in disjoint open sets. (ii) ⇒ (iii). Let A be a compact set and suppose that {x}rcl ∩ A = ∅. So for each a ∈ A, a /∈ {x}rcl by (ii) there exist disjoint open sets Ua and Va containing a and x, respectively. Thus the collection ν = {Ua : a ∈ A} is an open cover of the compact set A and so there exists a finite subcollection {Ua1, ..., Uan} of ν which covers A. Let U = ∪ n i=1Uai and V = ∩ n i=1Vai . Then U and V are disjoint open sets containing A and x, respectively. (iii) ⇒ (iv). Suppose that A and B are compact and {a}rcl ∩ B = ∅ for every a ∈ A. Then by (iii) for each a ∈ A there exist disjoint open sets Ua and Va containing a and B, respectively. The collection ν = {Ua : a ∈ A} is an open cover of the compact set A and so there exists a finite subcollection {Ua1, ..., Uan} of ν which covers A. Let U = ∪ n i=1Uai and V = ∩ n i=1Vai. Then U and V are disjoint open sets containing A and B, respectively. (iv) ⇒ (i). Suppose {x}rcl 6= {y}rcl. Then either x /∈ {y}rcl or y /∈ {x}rcl. For definiteness assume that y /∈ {x}rcl. Then {x}rcl ∩ {y}rcl = ∅ and so by (iv) there exist disjoint open sets U and V containing x and y, respectively. � c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 160 Rcl-spaces and closedness/completeness of certain function spaces Theorem 4.5. The disjoint topological sum of any family of Rcl-spaces is an Rcl-space. Theorem 4.6. The property of being an Rcl-space is closed under initial sources, i.e., the property of being an Rcl-space is an initial property. Proof. Let {fα : X → Yα : α ∈ Λ} be a family of functions, where each Yα is an Rcl-space and let X be equipped with initial topology. Let U be any open set in X and let x ∈ U. Then there exist α1, ..., αn ∈ Λ and open sets Vi ∈ Yαi(i = 1, ..., n) such that x ∈ f −1 α1 (V1) ∩ ... ∩ f −1 αn (Vn) ⊂ U. Since each Yα is an Rcl-space, there exists a cl-closed set Aαi in Yαi(i = 1, ..., n) such that fαi(x) ∈ Aαi ⊂ Vi. Since each fα is continuous, it follows that each f −1 αi (Aαi) is a cl-closed set in X. Let A = ∩n i=1f −1 αi (Aαi). Since any intersection of cl- closed sets is a cl-closed, A is a cl-closed set in X and x ∈ A ⊂ U so X is an Rcl-space. � As an immediate consequence of Theorem 4.6 we have the following. Theorem 4.7. The property of being an Rcl-space is hereditary, productive, sup-invariant, preimage invariant and projective4. Theorem 4.8. The category of Rcl-spaces and continuous maps is a full iso- morphism closed monoreflective as well as epireflective subcategory of TOP5. The following result gives a factorization of ultra Hausdorff property with Rcl-space as an essential ingredient. Theorem 4.9. Every ultra Hausdorff space is an Rcl-space. Conversely, every T0, Rcl-space is an ultra Hausdorff space. Proof. The first assertion is immediate, because in this case every singleton is cl-closed and so every open set is the union of cl-closed sets. Conversely, suppose that X is a T0, Rcl-space and let x, y ∈ X, x 6= y. By T0-property of X there exists an open set U containing one of the points x and y but not both. To be precise, assume that x ∈ U. Since X is an Rcl-space, there exists a cl-closed set A such that x ∈ A ⊂ U. Let A = ∩{Cα : α ∈ Λ}, where each Cα is a clopen set. Then there exists an α0 ∈ Λ such that y /∈ Cα0. Hence Cα0 and X \ Cα0 are disjoint clopen sets containing x and y, respectively and so X is an ultra Hausdorff space. � 5. Rcl-supercontinuous functions and Rcl-spaces Definition 5.1 ([37]). A function f : X → Y from a topological space X into a topological space Y is said to be Rcl-supercontinuous if for each x ∈ X and for each open set V containing f(x), there exists an rcl open set U containing x such that f(U) ⊂ V . 4A topological property P is said to be projective if whenever a product space has property P every co-ordinate space possesses property P. 5For the definition of categorical terms we refer the reader to Herrlich and Strecker [11]. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 161 J. K. Kohli and D. Singh It is immediate from the definition that every continuous function defined on an Rcl-space is Rcl-supercontinuous. Next we quote the following result from [37]. Theorem 5.2 ([37, Theorem 4.11]). Let f : X → ∏ α∈Λ Xα be defined by f(x) = (fα(x))α∈Λ, where fα : X → Xα is a function for each α ∈ Λ. Let∏ α∈Λ Xα be endowed with the product topology. Then f is Rcl-supercontinuous if and only if each fα is Rcl-supercontinuous. Now we give an alternative short proof of the following result from [37]. Theorem 5.3 ([37, Theorem 4.13]). Let f : X → Y be a function and g : X → X × Y be the graph function defined by g(x) = (x, f(x)) for each x ∈ X. Then g is Rcl-supercontinuous if and only if f is Rcl-supercontinuous and X is an Rcl-space. Proof. Observe that g = 1X ×f, where 1X denotes the identity function defined on X. Now by Theorem 5.2, g is Rcl-supercontinuous if and only if 1X and f both are Rcl-supercontinuous. Again 1X is Rcl-supercontinuous implies that each open set in X is rcl-open and so X is an Rcl-space. � Theorem 5.4. Let f : X → Y be an Rcl-supercontinuous open bijection. If either of the space X and Y is a T0-space, then X and Y are homeomorphic ultra Hausdorff spaces. Proof. By [37, Theorem 5.1] X and Y are homeomorphic Rcl-spaces. The last part of the theorem is immediate in view of the fact that a T0, Rcl-space is ultra Hausdorff (Theorem 4.9). � 6. Function spaces It is a well known fact that the function space C(X, Y ) of all continuous functions from a topological space X into a uniform space Y is not necessarily closed in Y X in the topology of pointwise convergence. However, it is closed in Y X in the topology of uniform convergence. It is of fundamental impor- tance in topology, analysis and several other branches of mathematics and its applications to know whether a given function space is closed / compact / complete in Y X or C(X, Y ) in the topology of pointwise convergence / uni- form convergence. Results of this nature and Ascoli type theorems abound in the literature (see [1, 12]). Sierpinski [29] showed that the set of all connected (Darboux) functions from a topological space X into a uniform space Y is not necessarily closed in Y X in the topology of uniform convergence. In contrast, Naimpally [25] showed that the set of all connectivity functions from a space X into a uniform space Y is closed in Y X in the topology of uniform convergence. Moreover, in [26] Naimpally introduced the notion of graph topology Γ for a function space and proved that the set of all almost continuous functions in the sense of Stalling [34] is not only closed in Y X in the graph topology but c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 162 Rcl-spaces and closedness/completeness of certain function spaces it represents the closure of C(X, Y ) in the graph topology. In the same vein, Hoyle [10] showed that the set SW(X, Y) of all somewhat continuous func- tions from a space X into a uniform space Y is closed in Y X in the topology of uniform convergence. Furthermore, Kohli and Aggarwal in [14] proved that the function space SC(X, Y ) of quasicontinuous ( ≡ semicontinuous) functions, Cα(X, Y ) of α-continuous functions, and L(X, Y) of cl-supercontinuous func- tions are closed in Y X in the topology of uniform convergence. In this section we strengthen the results of [14] and show that the set Rcl(X, Y ) ⊃ L(X, Y ) of all Rcl-supercontinuous functions is closed in Y X in the topology of uniform convergence. Definition 6.1. A subset A of a topological space X is said to be (i) semi open [22] (≡ quasi open [13]) if there exists an open set U in X such that U ⊂ A ⊂ U (ii) α-open [27] if A ⊂ (A0) 0 (iii) cl-open [32] if for each x ∈ A there exists a clopen set H such that x ∈ H ⊂ A. Definition 6.2. A function f : X → Y from a topological space X into a topological space Y is said to be a (i) connected (Darboux) function if f(A) is connected for every con- nected set A ⊂ X (ii) connectivity function if the graph of every connected subset of X is a connected subset of X × Y (iii) semicontinuous [22] (quasicontinuous [13]) if f−1(V ) is semi open in X for every open set V in Y (iv) α-continuous [24] if f−1(V ) is α-open in X for every open set V in Y (v) somewhat continuous [8] if for each open set V in Y such that f−1(V ) 6= ∅, then there exists a nonempty open set U in X such that U ⊂ f−1(V ), i.e. (f−1(V ))0 6= ∅. Remark 6.3. Somewhat continuous functions have also been referred to as fee- bly continuous (see [2, 6]) in the literature. However, Frolik [6] requires func- tions to be onto. We now recall the notion of the topology of uniform convergence. Let Y X = {f : X → Y is a function} be the set of all functions from a topological space X into a uniform space (Y, ν), where ν is a uniformity on Y . Let F ⊂ Y X. A basis for the uniformity of uniform convergence u for F is the collection {W(V ) : V ∈ ν}, where W(V ) = {(f, g) ∈ F × F : (f(x), g(x)) ∈ V for all x∈ X}. The uniform topology associated with u is called the topology of uni- form convergence. For details we refer the reader to [12]. Definition 6.4 ([12]). A uniform space (Y, ν) is said to be complete if and only if every Cauchy net in Y converges to a point in Y . c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 163 J. K. Kohli and D. Singh Theorem 6.5 ([12, p. 194]). A product of uniform spaces is complete if and only if each co-ordinate space is complete. Theorem 6.6. Let X be a topological space and let (Y, ν) be a uniform space. Then the set Rcl(X, Y ) of all Rcl-supercontinuous functions from X into Y is closed in Y X in the topology of uniform convergence. Further, if Y is a complete uniform space, then so is the function space Rcl(X, Y ) in the topology of uniform convergence. Proof. Let f ∈ Y X be the limit point of Rcl(X, Y ) which is not Rcl-supercontinuous at x ∈ X. Then there exists V ∈ ν such that f−1(V [f(x)]) does not con- tain any rcl-open set containing x. Choose a symmetric member W of ν such that WoWoW ⊂ V . Since f is a limit point of Rcl(X, Y ), there exists g ∈ Rcl(X, Y ) such that g(y) ∈ W [f(y)] for all y ∈ X. Then g ⊂ Wof and g−1 ⊂ f−1oW −1 = f−1oW and hence g−1oWog ⊂ f−1oWoWoWof ⊂ f−1oV of. Therefore g−1[W(g(x))] ⊂ f−1(V [f(x)]). Since f−1(V [f(x)]) does not contain any rcl-open set containing x, neither does g −1[W(g(x))] which contradicts Rcl-supercontinuity of g. Therefore f ∈ Rcl(X, Y ). The last asser- tion is immediate in view of Theorem 6.5 and the fact that a closed subspace of complete uniform space is complete. � Remark 6.7. In view of the above discussion we extend the following inclusions diagram from [14]. L(X, Y ) ⊂ Rcl(X, Y ) ⊂ C(X, Y ) ⊂ Cα(X, Y ) ⊂ SC(X, Y ) ⊂ SW(X, Y ) ⊂ Y X. Since in the topology of uniform convergence each of the above function space is a closed subspace of its succeeding one, the completeness of any one of them implies that of its predecessor. In particular, if Y is complete, then each of the above function space is complete. References [1] A. V. Arhangel’skii, General Topology III, Springer-Verlag, Berlin, 1995. [2] S. P. Arya and M. Deb, On mapping almost continuous in the sense of Froĺık, Math. Student 41 (1973), 311–321. [3] C. E. Aull, Functionally regular spaces, Indag. Math. 38 (1976), 281–288. [4] Á. Császár, General Topology, Adam Higler Ltd., Bristol, 1978. [5] A. S. Davis, Indexed system of neighbourhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886–893. [6] Z. Froĺık, Remarks concerning the invariance of Baire spaces under mapping, Czechoslo- vak Math. J. 11, no. 3 (1961), 381–385. [7] M. Ganster, On strongly s-regular spaces, Glasnik Mat. 25, no. 45 (1990), 195–201. [8] K. R. Gentry, and H. B. Hoyle, III, Somewhat continuous functions, Czechoslovak Math. J. 21, no. 1 (1971), 5–12. [9] N. C. Heldermann, Developability and some new regularity axioms, Can. J. Math. 33, no. 3 (1981), 641–663. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 164 Rcl-spaces and closedness/completeness of certain function spaces [10] H. B. Hoyle, III, Function spaces for somewhat continuous functions, Czechoslovak Math. J. 21, no. 1 (1971), 31–34. [11] H. Herrlich and G. E. Strecker, Category Theory An Introduction, Allyn and Bacon Inc. Bostan, 1973. [12] J. L. Kelly, General Topology, Van Nostrand, New York, 1955. [13] S. Kempisty, Sur les functions quasicontinuous, Fund. Math. 19 (1932), 184–197. [14] J. K. Kohli and J. Aggarwal, Closedness of certain classes of functions in the topology of uniform convergence, Demonstratio Math. 45, no. 4 (2012), 947–952. [15] J. K. Kohli and R. Kumar, z-supercontinuous functions, Indian J. Pure Appl. Math. 33, no. 7 (2002), 1097–1108. [16] J. K. Kohli and D. Singh, D-supercontinuous functions, Indian J. Pure Appl. Math. 32, no. 2 (2001), 227–235. [17] J. K. Kohli and D. Singh, Dδ-supercontinuous functions, Indian J. Pure Appl. Math. 34, no. 7 (2003), 1089–1100. [18] J. K. Kohli and D. Singh, Between regularity and complete regularity and a factorization of complete regularity, Studii Si Cercetari Seria Matematica 17 (2007), 125–134. [19] J. K. Kohli and D. Singh, Separation axioms between regular spaces and R0 spaces, preprint. [20] J. K. Kohli and D.Singh, Separation axioms between functionally regular spaces and R0 spaces, preprint. [21] J. K. Kohli, B. K. Tyagi, D. Singh and J. Aggarwal, Rδ-supercontinuous functions, Demonstratio Math. 47, no. 2 (2014), 433–448. [22] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 34–41. [23] J. Mack, Countable paracompactness and weak normality properties, Trans. Amer. Math. Soc. 148 (1970), 265–272. [24] A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, α-continuous and α-open mappings, Acta Math. Hungar. 41 1983, 213–218. [25] S. A. Naimpally, Function space topologies for connectivity and semiconnectivity func- tions, Canad. Math. Bull. 9 (1966), 349–352. [26] S. A. Naimpally, Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267–272. [27] O. Njástad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961–970. [28] N. A. Shanin, On separation in topological spaces, Dokl. Akad. Nauk SSSR, 38 (1943), 110–113. [29] W. Sierpiński, Sur une propriété de functions réelles quelconques, Matematiche (Cata- nia) 8 (1953), 43–48. [30] M. K. Singal and S. B. Niemse, z-continuous mappings, The Mathematics Student 66, no. 1-4 (1997), 193–210. [31] D. Singh, D∗-supercontinuous functions, Bull. Cal. Math. Soc. 94, no. 2 (2002), 67–76. [32] D. Singh, cl-supercontinuous functions, Appl. Gen. Topol. 8, no. 2 (2007), 293–300. [33] D. Singh, B. K. Tyagi, J. Aggarwal and J. K. Kohli, Rz-supercontinuous functions, Math. Bohemica, to appear. [34] J. R. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), 249–263. [35] R. Staum, The Algebra of bounded continuous functions into a nonarchimedean field, Pac. J. Math. 50, no. 1 (1974), 169–185. [36] L. A. Steen and J. A. Seebach, Jr., Counter Examples in Topology, Springer Verlag, New York, 1978. [37] B. K. Tyagi, J. K. Kohli and D. Singh, Rcl-supercontinuous functions, Demonstratio Math. 46, no. 1 (2013), 229–244. [38] R. Vaidyanathswamy, Treatise on Set Topology, Chelsa Publishing Company, New York, 1960. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 165 J. K. Kohli and D. Singh [39] W. T. Van East and H. Freudenthal, Trennung durch stetige Functionen in topologishen Raümen, Indag. Math. 15 (1951), 359–368. [40] N. K. Veličko, H-closed topological spaces, Amer. Math. Soc. Transl. 78, no. 2 (1968), 103–118. [41] G. J. Wong, On S-closed spaces, Acta Math. Sinica, 24 (1981), 55–63. [42] C. T. Yang, On paracompact spaces, Proc. Amer. Math. Soc. 5, no. 2 (1954), 185–194. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 166