() @ Appl. Gen. Topol. 18, no. 2 (2017), 331-344doi:10.4995/agt.2017.7149 c© AGT, UPV, 2017 A study of function space topologies for multifunctions Ankit Gupta a,∗ and Ratna Dev Sarma b, † a Department of Mathematics, University of Delhi, Delhi 110007, India (ankitsince1988@yahoo.co.in) b Department of Mathematics, Rajdhani College, University of Delhi, Delhi 110015, India. (ratna sarma@yahoo.com) Communicated by A. Tamariz-Mascarúa Abstract Function space topologies are investigated for the class of continuous multifunctions. Using the notion of continuous convergence, splitting- ness and admissibility are discussed for the topologies on continuous multifunctions. The theory of net of sets is further developed for this purpose. The (τ, µ)-topology on the class of continuous multifunctions is found to be upper admissible, while the compact-open topology is upper splitting. The point-open topology is the coarsest topology which is coordinately admissible, it is also the finest topology which is coor- dinately splitting. 2010 MSC: 54C35; 54A05; 54C60. Keywords: multifunction; function space topology; continuous convergence; splittingness; admissibility. 1. Introduction The interplay of properties of the topological spaces X and Y and those of the function space C(X, Y ) of continuous functions from X to Y has been an area of active research in topology. Several different sets of conditions under which the compact-open, Isbell or natural topologies on the set of continuous ∗Corresponding author. †This paper was prepared when the second author was on sabbatical leave. Received 24 January 2017 – Accepted 14 August 2017 http://dx.doi.org/10.4995/agt.2017.7149 A. Gupta and R. D. Sarma real-valued functions on a space may coincide, have been studied in [13]. A unified theory of function spaces and hyperspaces has been developed in [3]. In [4], it is shown that the intersection of all admissible topologies on C(X, Y ) is admissible under certain conditions. These and many other research papers published in the recent years are the testimony to the keen interest of the researchers in the study of function spaces. In [13], while discussing coincidence of the function space topologies, a natu- ral topology on the set of upper semi continuous set-valued functions has been constructed. Apart from this, the continuous multifunctions in the study of function spaces have been investigated by several researchers [9, 10, 12, 15, 20, 21, 22]. At the same time, the multifunctions are being extensively used now-a-days in several areas of mathematics such as Optimization theory, Frame theory, Approximation theory etc., to name a few. In this paper, we further develop the topological aspects of the function spaces for multifunctions. Starting from the basic level, we provide discussions for several new as well as already existing topologies for continuous multifunc- tions. We have adopted the net theoretic approach to discuss continuous con- vergence for the topology of multifunctions. The net theory for sets is further developed for the same purpose. Here it may be mentioned that characteriza- tions of upper semi-continuity and lower semi-continuity are provided in [14] using net convergence. In our paper, we show that under certain conditions, continuity of multifunctions implies a net-theoretic result which is similar to its counterpart for single-valued functions. Our study here is purely topolog- ical, unlike [11], where metric spaces and normed spaces are considered for similar results. Similarly, the continuous convergence introduced in our pa- per is different from that of [2] and [18]. In [2] and [18], upper and lower topologies, defined on the second space, are used for defining continuous con- vergence. However our definition is more straight forward and appears similar to its counterpart of single-valued functions. Conditions for splittingness (resp. upper and lower splittingness) and admissibility (resp. upper and lower ad- missibility) are obtained by using the concept of continuous convergence. The characterizations of admissibility and splittingness using net theory as shown in Arens and Dugundji [1] do not hold for multifunctions. Their variants are investigated in our paper. Several examples are provided to explain the intrin- sic differences between the topologies of continuous functions and topologies of continuous multifunctions. In the last section, several topologies on CM(Y, Z), the class of continuous multifunctions are studied in the light of splittingness and admissibility. While the (τ, µ)-topology is found to be upper admissible, the compact-open topology on CM(Y, Z) is upper splitting. The point-open topology is found to be the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting. During our investigation, we have also noticed that for a multifunction F : (X, τ) → (Y, µ) and U ∈ µ, the two different types of inverse images, that is, F +(U) and F −(U) types give rise to two families of open sets of τ. This leads to the possibility of having more than one dual topology for a given function c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 332 A study of function space topologies for multifunctions space topology on CM(Y, Z). However, further in depth research is required in this regard, which is beyond the scope of this paper. 2. Preliminaries Definition 2.1. A multifunction F : X → Y is a point-to-set correspondence from X to Y . We always assume that F(x) 6= ∅ for all x ∈ X. For each B ⊆ Y , F +(B) = {x ∈ X | F(x) ⊆ B} and F −(B) = {x ∈ X | F(x) ∩ B 6= ∅}. For each A ⊆ X, F(A) = ⋃ x∈A F(x). The collection of all the multifunctions from X to Y is denoted by Y XM. The following definitions and results are taken from the available literature. Definition 2.2. Let (X, τ) and (Y, µ) be two topological spaces. Then F : X → Y is called (i) upper semi continuous (or u.s.c., in brief) at x ∈ X if for each open set V ⊆ Y with F(x) ⊆ V , there exists an open set U of X such that x ∈ U and F(U) ⊆ V ; (ii) lower semi continuous (or l.s.c, in brief) at x ∈ X if for each open set V ⊆ Y with F(x) ∩ V 6= ∅, there exists an open set U of X such that x ∈ U and F(u) ∩ V 6= ∅ for every u ∈ U; (iii) continuous at x ∈ X, if it is both u.s.c. and l.s.c. at x; (iv) continuous (resp. u.s.c., l.s.c.) if it is continuous (resp. u.s.c., l.s.c.) at each point of X. If (X, τ) and (Y, µ) are two topological spaces and F : X → Y is a multi- function, then the following conditions are equivalent: (i) F is l.s.c. (resp. u.s.c.); (ii) F −(U) (resp. F +(U)) is open in X for each open subset U of Y ; (iii) F +(A) (resp. F −(A)) is closed in X for each closed subset A of Y . Definition 2.3. A multifunction F : X → Y is called a closed map if F(A) is closed in Y whenever A is closed in X. 3. A topology on CM(Y, Z) Now we proceed to define a topology on ZYM in the following way: Let (Y, τ) and (Z, µ) be two topological spaces. For U ∈ τ and V ∈ µ, we define (U, V ) = {F ∈ ZYM | F(U) ⊆ V } Let SMτ,µ = {(U, V ) | U ∈ τ, V ∈ µ}. Lemma 3.1. SMτ,µ is a subbasis for a topology on Z Y M. Proof. For F ∈ ZYM, we have F(∅) ⊆ V , for each V ∈ µ. Hence, we get F ∈ (∅, V ) for each V ∈ µ. Therefore ⋃ SMτ,µ = Z Y M. � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 333 A. Gupta and R. D. Sarma In our discussion, we take F = CM(Y, Z), the collection of all continuous multifunctions from Y to Z. The topology on F obtained in the above manner is denoted by TMτ,µ, and is called the (τ, µ)-topology on CM(Y, Z). This topology reduces to open-open topology τoo[17] if the multifunctions are replaced by functions. In fact, τoo turns out to be the relative topology of T M τ,µ, when we consider the subspace C(Y, Z) of CM(Y, Z). Lemma 3.1 ensures the existence of a topology on CM(Y, Z). In fact, several other interesting topologies exist on CM(Y, Z). We will discuss some of them in the last section. Definition 3.2. Let (Y, τ) and (Z, µ) be two topological spaces. Let (X, λ) be another topological space. For a multifunction G : X × Y → Z, we define a map G∗ : X → CM(Y, Z) by G ∗(x)(y) = G(x, y). The mappings G and G∗ related in this way are called associated maps. Definition 3.3. Let (Y, τ) and (Z, µ) be two topological spaces. A topology T on CM(Y, Z) is called (i) admissible (resp. upper admissible, lower admissible) if the evaluation mapping E : CM(Y, Z)× Y → Z defined by E(F, y) = F(y) is continuous (resp. u.s.c., l.s.c.). (ii) splitting (resp. upper splitting, lower splitting) if for each topological space X, continuity (resp. u.s.c., l.s.c.) of G : X × Y → Z implies the continuity of G∗ : X → CM(Y, Z), where G ∗ is the associated map of G. In [5], Georgiou, Iliadis and Papadopoulos have introduced one more vari- ation of admissibility and splittingness for function spaces. We extend those definitions for multifunctions as follow: Definition 3.4. Let (Y, τ) and (Z, µ) be two topological spaces. A topology T on CM(Y, Z) is called (i) coordinately admissible (resp. upper coordinately admissible, lower coor- dinately admissible) if the evaluation mapping E : CM(Y, Z) × Y → Z defined by E(F, y) = F(y) is coordinately continuous (resp. coordinately u.s.c., coordinately l.s.c.). (ii) coordinately splitting (resp. coordinately upper splitting, coordinately lower splitting) if for each topological space X, coordinately continuity (resp. coordinately u.s.c., coordinately l.s.c.) of G : X × Y → Z implies the continuity of G∗ : X → CM(Y, Z), where G ∗ is the associated map of G. Here, a map F : X × Y → Z is said to be coordinately continuous (resp. coordinately u.s.c., coordinately l.s.c.) if the maps Fx : Y → Z and Fy : X → Z defined by Fx(y) = F(x, y) and Fy(x) = F(x, y) are continuous (resp. u.s.c., l.s.c.) for every x ∈ X and for every y ∈ Y . The following two observations will be used at several places in the paper: (i) Let f : X → Y and G : Y → Z be a continuous function and a continuous (resp. u.s.c., l.s.c.) multifunction respectively. Then Gof is continuous (resp. u.s.c., l.s.c.). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 334 A study of function space topologies for multifunctions (ii) Let (Y, τ) and (Z, µ) be two topological spaces. A topology T on CM(Y, Z) is admissible (resp. upper admissible, lower admissible) if and only if for each topological space X, continuity of G∗ : X → CM(Y, Z) implies continuity (resp. u.s.c., l.s.c.) of G : X×Y → Z, where G is the associated map of G∗. 4. Continuous convergence of multifunctions In this section, we first investigate the relationships between the net conver- gence criteria and the continuity of multifunctions. Convergence of net of sets is required for this purpose. Definition 4.1 ([2, 16]). Let S = {An}n∈∆ be a net of sets in a topological space (X, τ). Then for any x ∈ X, we say (i) x ∈ Lim Inf(An) (or, x ∈ LI(An), in brief) if S eventually intersects every open neighbourhood of x, that is, given an open neighbourhood U of x, there exists m ∈ ∆, such that An ∩ U 6= ∅ for all n ≥ m. (ii) x ∈ Lim Sup(An) (or, x ∈ LS(An), in brief) if S frequently intersects every open neighbourhood of x, that is, given an open neighbourhood U of x and any m ∈ ∆, there exists an n ≥ m such that An ∩ U 6= ∅. (iii) The net of sets S = {An}n∈∆ is said to converge to A and we write Lim(An) = A if LI(An) = LS(An) = A. Lemma 4.2. For any net of sets, we have LI(An) ⊆ LS(An). Theorem 4.3. Let F be a multifunction from a topological space (X, τ) to a regular topological space (Y, µ). Let {xn}n∈∆ be a net in X, which converges to x in X. Then the net {F(xn)}n∈∆ converges to F(x), if F is continuous at x ∈ X and F(x) is closed in (Y, µ). Proof. Let y ∈ F(x) and V be any open neighbourhood of y in Y . Then V ∩ F(x) 6= ∅. Thus x ∈ F −(V ). As F is continuous, F −(V ) is open. Therefore, there exists an open neighbourhood U of x such that x ∈ U ⊆ F −(V ). Since {xn}n∈∆ converges to x, we have xn ∈ U ⊆ F −(V ) eventually. Therefore, F(xn) ∩ V 6= ∅ eventually and hence y ∈ LI(F(xn)n∈∆). Thus, we have F(x) ⊆ LI (F(xn)n∈∆). Now we claim that LS(F(xn)n∈∆) ⊆ F(x). Let y /∈ F(x). Since (Y, µ) is regular and F(x) is a closed set not containing y, therefore, there exist disjoint open sets U and V such that y ∈ U and F(x) ⊆ V . As multifunction F is given to be continuous at x ∈ X, there exists an open neighbourhood W of x such that F(W) ⊆ V . Again, since the given net {xn}n∈∆ converges to x, therefore xn ∈ W eventually. Then we have F(xn) ⊆ V for all n ≥ n0, for some n0 ∈ ∆. This implies that F(xn) ∩ U = ∅ for all n ≥ n0. Hence, y /∈ LS(F(xn)n∈∆). Thus, we have LS(F(xn)n∈∆) ⊆ F(x). Hence, F(x) = LI(F(xn)n∈∆) = LS(F(xn)n∈∆). Therefore {F(xn)}n∈∆ converges to F(x). � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 335 A. Gupta and R. D. Sarma In the above theorem, the condition of regularity of the space (Y, µ) and closedness of F(x) can not be relaxed. Here we provide examples to demon- strate this. Example 4.4. Let X = R be the set of real numbers with the usual topology U and µ be the irrational slope topology defined on Y = {(x, y) | y ≥ 0, x, y ∈ Q}. We fix some irrational number θ. The irrational slope topol- ogy[23] µ on Y is generated by neighbourhoods of the form Nǫ((x, y)) = {(x, y)} ∪ Bǫ(x + y/θ) ∪ Bǫ(x − y/θ) where Bǫ(a) = {(r, 0) ∈ Y | |r − a| < ǫ} is the collection of all rationals in (a − ǫ, a + ǫ). This irrational slope topology is Hausdorff but not regular[23]. Let F : (X, U) → (Y, µ) be a multifunction defined by F(x) =        {(1, 5)}, x = 1 Bǫ(1 − 5/θ), for all x ∈ (0, 1) Bǫ(1 + 5/θ), for all x ∈ (1, 2) {(5, 10)} otherwise. We claim that (i) F(1) = {(1, 5)} is closed in (Y, µ); (ii) F is continuous at x = 1; (iii) {F(1 − 1/n)}n∈N does not converge to F(1) in (Y, µ), while {1 − 1/n}n∈N converges to 1 in (X, U). Below we provide justifications for claim (i), (ii) and (iii): (i) Since (Y, τ) is a Hausdorff space, there does not exist any y ∈ Y distinct from the point (1, 5) such that every neighbourhood U of y intersects (1, 5). Thus cl{(1, 5)} = {(1, 5)}. Thus F(1) = {(1, 5)} is closed; (ii) First we show that F is u.s.c. at x = 1. Let V be any open set in Y such that F(1) ⊆ V . Since V is open in Y , we have {(1, 5)} ∪ Bǫ(1 + 5/θ) ∪ Bǫ(1 − 5/θ) ⊆ V , for some ǫ > 0. Therefore F ((0, 2)) ⊆ V , where (0, 2) is neighbourhood of 1. Hence F is u.s.c. at x = 1. Now, we show that F is l.s.c. at x = 1. Let V be any open set in Y such that F(1) ∩ V 6= ∅. Any open set containing (1, 5) must contain Bǫ(1 + 5/θ) ∪ Bǫ(1 − 5/θ) also. Thus, again we have F(x) ∩ V 6= ∅ for all x ∈ (0, 2). Therefore F is l.s.c. at x = 1. Hence F is continuous at x = 1; (iii) Next, we prove that {F(1 − 1/n)}n∈N does not converge to F(1), that is F(1) 6= LI(F(xn)), where xn = 1 − 1/n. Let y ∈ Bǫ(1−5/θ). Then every neighbourhood of y intersects F(1−1/n) for all n. Thus y ∈ LI(F(xn)). But y /∈ F(1). Hence {F(1 − 1/n)}n∈N does not converge to F(1) in (Y, µ) while {1 − 1/n}n∈N converges to 1 in (X, U). In the next example, we show that the condition of closedness can not be relaxed. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 336 A study of function space topologies for multifunctions Example 4.5. Let X = R be the set of real numbers with the usual topol- ogy U and let Y = {a, b, c, d} with topology τ = {∅, Y, {a, b}, {c, d}}. Let F : (X, U) → (Y, τ) be a multifunction defined by F(x) =    {a} for x = 1, {a, b} for all x ∈ (1 − 1/n, 1) ∪ (1, 1 + 1/n) for all n ≥ m, for some m ∈ N, {c} otherwise. Here (Y, τ) is regular and F(1) = {a} is not closed in Y . Now, we claim that (i) F is continuous at x = 1; (ii) {F(1-1/n)}n∈N does not converge to F(1) in (Y, τ), while {1-1/n}n∈N converges to 1 in (X, U). We have (i) First we show that F is u.s.c. at x = 1. Let V be any open set in Y such that F(1) ⊆ V . Since V is open in Y , we have {a, b} ⊆ V . Therefore there exists (1-1/m, 1 + 1/m), an open neighbourhood of 1 in (X, U) such that F((1-1/m, 1 + 1/m)) ⊆ V . Hence F is u.s.c. at x = 1. Now, we show that F is l.s.c. at x = 1. Let V be any open set in Y such that F(1) ∩ V 6= ∅. Any open set con- taining {a} must contains {a, b} also. Thus, again we have F(x)∩V 6= ∅ for all x ∈ (1 − 1/m, 1 + 1/m). Therefore F is l.s.c. at x = 1. Hence F is continuous at x = 1. (ii) Now, we show that {F(1-1/n)}n∈N does not converge to F(1), that is F(1) 6= LI(F(xn)), where xn = 1-1/n. Let y = b, then every neighbourhood of y intersects F(1 − 1/n) for all n ≥ m. Thus y ∈ LI(F(xn)). But y /∈ F(1). Hence {F(1 − 1/n)}n∈N does not converge to F(1) in (Y, τ), while {1 − 1/n}n∈N converges to 1 in (X, U). The condition which we imposed on F in Theorem 4.3, that is, F(x) is closed for every netwise limit x ∈ X can not be replaced by the condition that F is a closed map. (Here, x is a netwise limit means x is a limit of some non-trivial net in X.) For this, first we define a topology on R, which is the Countable Complement Extension Topology [23], as given below. Definition 4.6. Let X = R be the set of real numbers and τ1 be the Euclidean topology and τ2 be the topology of countable complements on X. We define τ to be the smallest topology generated by τ1 ∪ τ2. Here a set U is open in τ if and only if U = O�A, where O ∈ τ1 and A is countable. Then τ is the Countable Complement Extension Topology on R. Example 4.7. Let X and Y be the set of real numbers having the co-countable topology and the countable complement extension topology respectively. Let F : X → Y be a multifunction defined as F(x) = (−∞, −2 − 1/x] ∪ {x} for all c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 337 A. Gupta and R. D. Sarma x ∈ X. Here F(N) = (−∞, −2) ∪ N, which is not a closed set in Y . Hence F is not a closed map, although F(x) is closed for all x ∈ X. The following result available in [14], may be treated as a partial converse of Theorem 4.3. Theorem 4.8. Let (X, τ) and (Y, µ) be two topological spaces. Let F : X → Y be a multifunction. Then F is lower semi continuous at x ∈ X if for any net {xn}n∈∆ in X converging to x ∈ X, the image net {F(xn)}n∈∆ converges to F(x). Here we also mention to our readers that in [14], it is proved that if (X, τ) is a compact Hausdroff space and {An}n∈∆ is a net of sets, then Lim(An) = A if and only if {An}n∈∆ converges to A under Vietoris Topology τV of X. Generalized nets, defined in [19], were used in [7] to introduce the notion of continuous convergence for function spaces on generalized topologies. Here, we use net theory to the introduce the concept of continuous convergence for multifunctions. We shall use this concept extensively in our paper for classify- ing various topologies on CM(Y, Z). However, before coming to that, below we provide a small discussion related to directed sets and convergence of nets. Let ∆ be a directed set. We add a point ∞ to ∆ satisfying ∞ ≥ n for all n ∈ ∆ and write ∆0 = ∆ ∪ {∞}. Then a topology τ0 may be generated on ∆0 by declaring every singleton of ∆ is open and neighbourhood of ∞ to be all the sets of the form Un0 = {n : n ≥ n0}, where n0 is any arbitrary member of ∆. Lemma 4.9 ([1]). Let (Y, µ) be a topological space and {yn}n∈∆ be a net in Y . Then {yn}n∈∆ converges to y if and only if the function s : ∆0 → Y defined by s(n) = yn for n ∈ ∆ and s(∞) = y is continuous at ∞. In the next set of theorems, we will provide some characterizations of split- tingness and admissibility of topologies on CM(Y, Z). First we define continu- ous convergence for multifunctions. Definition 4.10. Let {Fn}n∈∆ be a net in CM(Y, Z). Then {Fn}n∈∆ is said to continuously converge to F if for each net {ym}m∈σ in Y converging to y, {Fn(ym)}(n,m)∈∆×σ converges to F(y) in Z. If we take functions in place of multifunctions, the above definition coincides with that of continuous convergence of functions defined in [1]. Theorem 4.11. Let (Y, τ) and (Z, µ) be two topological spaces in which (Z, µ) is regular. Let T be a topology on CM(Y, Z) such that for each net {Fn}n∈∆ in CM(Y, Z), continuous convergence of {Fn}n∈∆ to F implies {Fn}n∈∆ converges to F under T, provided G(x, y) is closed for every continuous map G : X ×Y → Z and for every netwise limit (x, y) ∈ X × Y . Then T is splitting. Proof. Suppose, continuous convergence implies convergence. Let G : X ×Y → Z be continuous. Let {xn}n∈∆ be a convergent net which converges to x in X. We need to show that {G∗(xn)}n∈∆ converges to G ∗(x) in CM(Y, Z). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 338 A study of function space topologies for multifunctions Let {ym}m∈σ be a net in Y converging to y. Then {(xn, ym)}(n,m)∈∆×σ con- verges to (x, y) in X × Y . Hence {G(xn, ym)} converges to G(x, y), in view of Theorem 4.3 because G is continuous and Z is regular. Let us define that G∗(xn) = Fn and G ∗(x) = F . Then G(xn, ym) = G ∗(xn)(ym) = Fn(ym) and G(x, y) = G∗(x)(y) = F(y). That is, {Fn(ym)}(n,m)∈∆×σ converges to F(y). Thus {Fn}n∈∆ continuously converges to F . Therefore by the given condition {Fn}n∈∆ converges to F in CM(Y, Z). That is, {G ∗(xn)}n∈∆ converges to G∗(x). Hence G∗ is continuous, that is, T is splitting. � Our next result provides a partial converse of the above theorem. Theorem 4.12. Let (Y, τ) and (Z, µ) be two topological spaces such that T on CM(Y, Z) is lower splitting. Then for each net {Fn}n∈∆ in CM(Y, Z), con- tinuous convergence of {Fn}n∈∆ to F implies {Fn}n∈∆ converges to F under T. Proof. Let T be lower splitting and {Fn}n∈∆ converge continuously to F . Let ∆0 = ∆ ∪ {∞} be the topological space generated from ∆. We define G : ∆0 × Y → Z by G(n, y) = Fn(y) and G(∞, y) = F(y). Now, we claim that G is l.s.c.. Let W be an open set in Z such that G(n, y) ∩ W 6= ∅, that is, Fn(y) ∩ W 6= ∅. Since Fn ∈ CM(Y, Z), therefore there exists an open neighbourhood V of y such that Fn(v)∩W 6= ∅ for each v ∈ V . Thus, we have G(n, v) ∩ W 6= ∅ for all (n, v) ∈ n × V . Therefore G is l.s.c. at (n, y). Now, we show that G is l.s.c. at (∞, y). It is clear that the only non-constant net in ∆0 is {n}, which converges to ∞. Hence if S is a convergent net in ∆0 × Y , then we have, S = S1 × S2, where S1 = {n} and S2 = {ym}m∈σ is some convergent net in Y . Then S converges to {∞}×{y}, for some y ∈ Y such that {ym}m∈σ converges to y. Then we have, G(S) = {Fn(ym)}(n,m)∈∆×σ. By continuous convergence of {Fn}, G(S) converges to F(y) = G(∞, y). Hence G is lower semi continuous at (∞, y). Thus G is l.s.c. on ∆0 × Y . As T is lower splitting, this implies that G∗ is continuous. As {n} converges to ∞ in ∆0, we have , {G∗(n)}n∈∆ converges to G ∗(∞). Now, G∗(n)(y) = G(n, y) = Fn(y) and G∗(∞)(y) = G(∞, y) = F(y). That is, G∗(n) = Fn and G ∗(∞) = F , hence {Fn} converges to F in CM(Y, Z). � Remark 4.13. If we take the subfamily C(Y, Z), instead of CM(Y, Z), then Theorem 4.11 and Theorem 4.12 reduce to the following, which was proved in [1]. Corollary 4.14. Let (Y, µ) and (Z, τ) be two topological spaces. A topology T in C(Y, Z) is splitting if and only if for any net {fn}n∈∆ in C(Y, Z), continuous convergence of {fn}n∈∆ to f implies convergence of {fn}n∈∆ to f in T. Proof. It holds in view of the fact that net-theoretic characterization of conti- nuity of functions does not need regularity, nor closedness of f(x). � In the following two results, we investigate the relationship between admis- sibility and continuous convergence. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 339 A. Gupta and R. D. Sarma Theorem 4.15. Let (Y, τ) and (Z, µ) be two topological spaces. A topology T on CM(Y, Z) is lower admissible if for any net {Fn}n∈∆ in CM(Y, Z), {Fn}n∈∆ converges to F in T implies continuous convergence of {Fn}n∈∆ to F. Proof. Let G∗ : X → CM(Y, Z) be continuous. We have to show that the asso- ciated map G is lower semi continuous. Let {xn, yn}n∈∆ be a convergent net which converges to (x, y) ∈ X × Y . Then {xn}n∈∆ converges to x in X and {yn}n∈∆ converges to y in Y . Since {xn}n∈∆ converges to x and G ∗ is contin- uous, therefore {G∗(xn)}n∈∆ converges to G ∗(x), that is, {Fxn}n∈∆ converges to Fx in CM(Y, Z), where Fxn = G ∗(xn) and Fx = G ∗(x) respectively. Then, by the given hypothesis, {Fxn}n∈∆ continuously converges to Fx. Hence for the convergent net {yn}n∈∆ which converges to y in Y , we have {Fxn(yn)}n∈∆ converges to Fx(y), that is, {G(xn, yn)}n∈∆ converges to G(x, y). Hence G is lower semi continuous. Therefore T is lower admissible. � For the converse part, we have the following result: Theorem 4.16. Let (Y, τ) and (Z, µ) be two topological spaces in which (Z, µ) is regular. Suppose a topology T on CM(Y, Z) is admissible. Then for each net {Fn}n∈∆ in CM(Y, Z), convergence of {Fn}n∈∆ to F implies {Fn}n∈∆ con- tinuously converges to F under T, provided G(x, y) is closed for every netwise limit (x, y) ∈ X × Y , for every map G : X × Y → Z and for any topological space (X, λ). Proof. Let T be admissible and {ym}m∈σ be any net in Y such that {ym}m∈σ converges to y in Y . Let {Fn}n∈∆ be any net in CM(Y, Z) such that {Fn}n∈∆ converges to F . Let us define G∗ : ∆0 → CM(Y, Z) as G ∗(n) = Fn and G∗(∞) = F , where ∆0 is generated by ∆. Now the only non-constant net in ∆0 is {n} which converges to ∞. Also, {G ∗(n)}n∈∆ = {Fn}n∈∆ converges to F = G∗(∞). Hence, G∗ is continuous. Therefore G : ∆0 ×Y → Z is continuous as T is admissible. Now {n, ym}(n,m)∈∆×σ is a convergent net in ∆0 × Y which converges to (∞, y). Therefore, in view of Theorem 4.3, {G(n, ym)}(n,m)∈∆×σ converges to G(∞, y), that is, {G∗(n)(ym)}(n,m)∈∆×σ converges to G ∗(∞)(y). This implies {Fn(ym)}(n,m)∈∆×σ converges to F(y). Hence {Fn} continuously converges to F . � As a corollary of Theorem 4.15 and Theorem 4.16, we get the following result for C(Y, Z) [1]: Corollary 4.17. Let (Y, τ) and (Z, µ) be two topological spaces. A topology T on C(Y, Z) is admissible if and only if for any net {fn}n∈∆ in C(Y, Z), {fn}n∈∆ converges to f in T implies continuous convergence of {fn}n∈∆ to f. In [1], Arens and Dugundji provided few lemmas for function spaces which are valid for function spaces for multifunctions as well. Below we mention them without proof. Here µ ≥ τ, means τ ⊆ µ. Lemma 4.18. Let τ and µ be two topologies on CM(Y, Z). If τ is admissible (resp. upper admissible, lower admissible) and µ ≥ τ, then µ is admissible c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 340 A study of function space topologies for multifunctions (resp. upper admissible, lower admissible). If µ is splitting (resp. upper split- ting, lower splitting) and µ ≥ τ, then τ is splitting (resp. upper splitting, lower splitting). Lemma 4.19. If µ is splitting (resp. upper splitting, lower splitting) and τ is admissible (resp. upper admissible, lower admissible) on CM(Y, Z), then µ ≤ τ. The following theorem, provided by Georgiou, Iliadis and Papadopoulos in [5] for function space is also valid for the function spaces for multifunctions. That is, Theorem 4.20. The following hold good: (i) Every coordinately splitting (resp. coordinately upper splitting, coordi- nately lower splitting) topology on CM(Y, Z) is splitting (resp. upper splitting, lower splitting). (ii) Every admissible (resp. upper admissible, lower admissible) topology on CM(Y, Z) is coordinately admissible (resp. coordinately upper admissible, coordinately lower admissible). 5. Various topologies over CM(Y, Z) In Section 3, we introduce the (τ, µ)-topology, denoted by TMτ,µ, on CM(Y, Z). In this section, we study several other topologies over CM(Y, Z) in the light of splittingness and admissibility. We also study the interrelationship between these topologies. First, we come back to the topology TMτ,µ on CM(Y, Z) defined in Section 3. We show that TMτ,µ is upper admissible on CM(Y, Z). Theorem 5.1. Let (Y, τ) and (Z, µ) be two topological spaces. Then the topol- ogy TMτ,µ is upper admissible for CM(Y, Z). Proof. Let (F, y) ∈ CM(Y, Z) × Y and let V ∈ µ such that E(F, y) ⊆ V . This implies F(y) ⊆ V . Since F is continuous, there exists an open set W containing y such that F(W) ⊆ V , that is, F ∈ (W, V ). Thus E((W, V ) × W) ⊆ V . Thus the evaluation map E is u.s.c.. Hence CM(Y, Z) is upper admissible. � Below we provide definitions for the compact-open topology and the point- open topology for multifunctions, and investigate some of their properties. Let (Y, τ) and (Z, µ) be two topological spaces. Then we define (C, V ) = {F ∈ CM(Y, Z) | F(C) ⊆ V } (y, V ) = {F ∈ CM(Y, Z) | F(y) ⊆ V } where C is a compact subset of Y , y ∈ Y and V ∈ µ. Let SMco = {(C, V ) | C is compact in Y and V ∈ µ} and S M po = {(y, V ) | y ∈ Y, V ∈ µ}. It can be shown that SMco and S M po form subbasis for two topologies on CM(Y, Z). These topologies are called the compact-open topology and the point-open topol- ogy respectively. They are denoted by TMco and T M po respectively. Clearly T M co is finer than TMpo . For further study on compact-open topology and compact convergence, one may refer to [9, 10, 12]. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 341 A. Gupta and R. D. Sarma Theorem 5.2. The compact-open topology over CM(Y, Z) is upper splitting. Proof. Left for the readers. � In our next result, we prove that the point-open topology for multifunctions is coordinately admissible. Theorem 5.3. Let (Y, τ) and (Z, µ) be two topological spaces. Then the topol- ogy TMpo on CM(Y, Z) is coordinately admissible. Also, this is the coarsest topology on CM(Y, Z) which is coordinately admissible. Proof. First we show that TMpo is coordinately upper admissible. For this, let F ∈ CM(Y, Z) and V ∈ µ such that Ey(F) ⊆ V . This implies F(y) ⊆ V . Consider, (y, V ) ∈ TMpo . We have Ey((y, V )) ⊆ V . Hence the evaluation map Ey is coordinately u.s.c. Now, let y ∈ Y and V ∈ µ such that EF (y) ⊆ V . This implies F(y) ⊆ V . Since F ∈ CM(Y, Z), then there exists an open neighbourhood W of y such that F(W) ⊆ V , then we have EF (w) ⊆ V for every w ∈ W . Hence the evaluation map EF is coordinately u.s.c. Therefore, T M po is coordinately upper admissible. Now, consider F ∈ CM(Y, Z) and V ∈ µ such that Ey(F) ∩ V 6= ∅, that is, F(y) ∩ V 6= ∅. Then again, we have (y, V ) ∈ TMpo such that Ey(F) ∩ V 6= ∅ for each F ∈ (y, V ). Hence the evaluation map Ey is coordinately l.s.c. Similarly, we can show that EF is coordinately l.s.c.. Thus T M po is coordinately lower admissible as well. Accordingly, TMpo is coordinately admissible. Now, we show that it is the coarsest topology on CM(Y, Z) having this property. Let (y, U) be a subbasic open set in TMpo and U be another topology which is coordinately admissible. We show that (y, U) is open in U. Consider F ∈ (y, U), that is, F(y) ⊆ U. Therefore, Ey(F) ⊆ U. Since the topology U is coordinately admissible, the evaluation map Ey is continuous. Therefore there exists an open set V in U containing F such that Ey(V ) ⊆ U. That is, F ∈ V ⊆ (y, U). Hence (y, U) is open in U. � Our next result shows that TMpo is coordinately upper splitting and hence upper splitting. Theorem 5.4. Let (Y, τ) and (Z, µ) be two topological spaces. Then the topol- ogy TMpo is coordinately upper splitting. Proof. Let (X, λ) be any topological space such that G : X × Y → Z is coor- dinately upper semi continuous. We have to show that its associated mapping G∗ : X → CM(Y, Z) is continuous. Let x ∈ X and (y, V ) be a subbasic open set in TMpo with G ∗(x) ∈ (y, V ). That is, G∗(x)(y) ⊆ V . Then, G(x, y) ⊆ V . Since the map G : X × Y → Z is coordinately upper semi continuous, there- fore Gy : X → Z is upper semi continuous for each y. Hence there exists an open neighbourhood Ox of x such that Gy(Ox) ⊆ V . Consequently, we have c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 342 A study of function space topologies for multifunctions G∗(Ox) ⊆ (y, V ). Hence G ∗ is continuous. Accordingly TMpo is coordinately upper splitting. � In view of Lemma 4.18 and 4.19, the above two results lead us to the following interesting result: Theorem 5.5. The point-open topology TMpo over CM(Y, Z) is the unique topol- ogy which is coordinately upper splitting as well as coordinately upper admis- sible. It is also the finest coordinately upper splitting as well as the coarsest coordinately upper admissible topology on CM(Y, Z). The Corollary 3.6 of [5] on C(Y, Z) can be viewed as a particular case of the above result. Remark 5.6. The open sets of the domain space which can be realized as pre- images of continuous functions have been used to define the dual topologies for a given function space topology in [6] and [8]. Several interesting relation- ships are established between a function space topology and its dual topology in these studies. Investigations in this regard may also be carried out for multifunction as well. However, in case of multifunctions, the development is not straightforward. It is due to the fact that, for a continuous multifunction F : (X, τ) → (Y, µ), the inverse images of open sets form two different classes in (X, τ) formed by F +(U) and F −(U) types of sets where U ∈ µ. A detailed discussion about the same is beyond the scope of the present paper and needs further investigation. References [1] R. Arens and J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951), 5–31. [2] J. Cao, I. L. Reilly and M. V. Vamanamurthy, Comparison of convergences for multi- functions, Demonstratio Math. 30 (1997), 171–182. [3] S. Dolecki and F. Mynard, A unified theory of function spaces and hyperspaces: local properties, Houston J. Math. 40, no. 1 (2014), 285–318. [4] D. N. Georgiou and S. D. Iliadis, On the greatest splitting topology, Topology Appl. 156 (2008), 70–75. [5] D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos, Topology on function spaces and the coordinate continuity, Topology Proc. 25 (2000), 507–517. [6] D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos, On dual topologies, Topology Appl. 140 (2004), 57–68. [7] A. Gupta and R. D. Sarma, Function space topologies for generalized topological spaces, J. Adv. Res. Pure Math. 7, no. 4 (2015), 103–112. [8] A. Gupta and R. D. Sarma, On dual topologies concerning function spaces over Cµ,ν(Y, Z), preprint. [9] V. G. Gupta, Compact convergence for multifunctions, Pure Appl. Math. Sci. 17 (1983), 35–40. [10] V. G. Gupta, Compact convergence topology for multi-valued functions, Proc. Nat. Acad. Sci. India Sect. A 53 (1983), 164–167. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 343 A. Gupta and R. D. Sarma [11] S. Hu and N. S. Papageorgiou, Handbook of multivalued analysis, Vol. I Theory, Kluwer Academic Publishers, Dordrecht, 1997. [12] P. Jain and S. P. Arya, Some function space topologies for multifunctions, India J. Pure Appl. Math. 6 (1975), 1488–1506. [13] F. Jordan, Coincidence of function space topologies, Topology Appl. 157 (2010), 336– 351. [14] E. Klein and A. Thompson, Theory of correspondences: including applications to mathe- matical economics, Canadian Mathematical Society Series of Monographs and Advanced texts. J. Wiley & Sons, 1984. [15] V. J. Mancuso, An Ascoli theorem for multi-valued functions, J. Austral. Math. Soc. 12 (1971), 466–472. [16] S. Mrowka, On Convergence of nets of sets, Fund. Math. 45 (1958) 237–246. [17] K. Porter, The open-open topology for function spaces, Inter. J. Math. and Math. Sci. 12 (1993), 111–116. [18] M. Przemski, On continuous convergence for nets of multifunctions, Demonstratio Math. 44 (2011), 181–200. [19] R. D. Sarma, On convergence in generalized topology, Int. J. Pure Appl. Math. 60, no. 2 (2010), 205–210. [20] R. E. Smithson, Topologies on sets of relations, J. Natur. Sci. and Math. 11 (1971), 43–50. [21] R. E. Smithson, Uniform convergence for multifunctions, Pacific J. Math. 39 (1971), 253–259. [22] R. E. Smithson, Multifunctions, Nieuw. Arch. Wisk. 20, no. 3 (1972), 31–53. [23] L. A. Steen and J. A. Seebach, Counterexamples in topology, Springer, New York 1978. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 344