Graph topologies on closed multifunctions

@ Applied General Topology c Universidad Polit´ecnica de Valencia Volume 4, No. 2, 2003 pp. 445–465 Graph topologies on closed multifunctions Giuseppe Di Maio, Enrico Meccariello and Somashekhar Naimpally Dedicated by the first two authors to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. In this paper we study function space topologies on closed multifunctions, i.e. closed relations on X×Yusing various hypertopologies. The hypertopologies are in essence, graph topologies i.e topologies on functions considered as graphs which are subsets of X×Y. We also study several topologies, including one that is derived from the Attouch-Wets filter on the range. We state embedding theorems which enable us to generalize and prove some recent results in the literature with the use of known results in the hyperspace of the range space and in the function space topologies of ordinary functions. 2000 AMS Classification: 54B20, 54C25, 54C35, 54C60, 54E05, 54E15. Keywords: hyperspaces, function spaces, graph topologies, Vietoris topology, Fell topology, uniform convergence on compacta, U-topology, ∆-topology, proximal ∆-topology, ∆U-topology, proximal ∆U-topology, Hausdorff-Bourbaki uniformity, ∆-Attouch-Wets filter. 1. Introduction. Recently McCoy [24] studied relations among four hyperspace topologies (viz. Fell topology, Fell uniform topology, Vietoris topology and HausdorffBourbaki topology) and the corresponding topologies on set valued maps. In this paper we plan to study the subject comprehensively in more general situations. We recall that for a topological space Zthe hyperspace, 2Z, of closed subsets of Zhas a number of natural topologies on it obtained from the topology on Z. In our setting (X, τ1) and (Y, τ2) denote Hausdorff topological spaces and Zthe product space X×Yequipped with the product topology τ=τ1×τ2. If δ1and δ2are compatible proximities on Xand Yrespectively, then on Zis assigned the product proximity δ=δ1×δ2. The hyperspace 2Z= 2X×Ycan be considered as the space Fof all set valued maps on Xto 2Ytaking points 446 G. Di Maio, E. Meccariello and S. Naimpally of Xto (possibly empty) closed subsets of Y. We do not distinguish between a function f∈Fand its graph {(x, f(x)) : x∈X} ⊂ Z=X×Y. Thus our study includes topologies on the spaces of partial maps studied first in 1936 and which are being studied intensively in recent times ([1], [2], [3], [13], [19], [20], [23], [27], [33], [34]). Given a Hausdorff topological space Z, for each subset Eof Z,clZE,intE and Ecstand for the closure, interior and complement of Ein Z. Moreover E−={A∈2Z:A∩E6=∅}; E+={A∈2Z:A⊂E}. Futhermore, if δis a compatible proximity on Z(for details see [30]), we set E++ δ={A∈2Z:AδE}. (Note: AδEiff A6δEcwhere 6δdenotes the negation of δ). We omit δif it is clear from the context and write E++ δsimply as E++. We recall that the set of all compatible proximities on Zis partially ordered as follows: δ1≤δ2iff whenever A,B⊂Zand A6δ1B, then A6δ2B(see [30]). Some special cases of δare: δ0the fine LO-proxmity on Zgiven by Aδ0Biff clZA∩clZB6=∅. δ0is called the Wallman proximity. It is well known that δ0is, by far, the most important compatible LOproximity on Z, and that δ0is EF iff Zis normal (Urysohn’s Lemma). If Zis Tychonoff and Vis a compatible uniformity on Z, then δ(V) denotes the EF -proximity on Zgiven by Aδ(V)Biff V(A)∩B6=∅for each V∈ V. δ(V) is called the uniform proximity (induced by V). If Zis a metrizable space with metric d, then δ(d) is the EF -proximity on Zgiven by Aδ(d)Biff Dd(A, B) = inf{d(a, b) : a∈A, b ∈B}= 0. δ(d) is called the metric proximity (induced by d). For Z=X×Y, we use the symbol ∆ (resp. ∆1, ∆2) to denote a subfamily of CL(Z) = 2Z\ {∅}(resp. of CL(X) = 2X\ {∅},CL(Y) = 2Y\ {∅}) which is a cobase, i.e. (a) is closed under finite unions; and (b) contains the singletons. Acover is a cobase which is also closed hereditary. Moreover, we assume (c) ∆1×∆2⊂∆. In some cases, in addition to the above condition, we suppose Graph topologies on closed multifunctions 447 (d) p1(∆) ⊂∆1and p2(∆) ⊂∆2, where p1and p2are projections from Z to Xand Yrespectively. A typical and important example of a cover is ∆ = K(Z), the family of all nonempty compact subsets of Z,∆1=K(X),∆2=K(Y). Moreover, in this case (c)−(d)also hold. In what follows, unless explicitly stated, we assume always that ∆ ⊂CL(Z) (resp. ∆1⊂CL(X), ∆2⊂CL(Y)) is a cobase. We now describe some hypertopologies on 2Z(for details see [4]). Suppose δis a compatible LO-proximity on Z. The lower Vietoris topology τ− Von 2Zhas a subbase {W−:W∈τ}. The upper ∆-topology τ(∆)+on 2Zhas a base {W+:Wc∈∆}. The upper proximal ∆-topology σ(∆, δ)+on 2Zhas a base {W++ :Wc∈∆}. The ∆ topology τ(∆) on 2Zequals τ− V∨τ(∆)+. The proximal ∆topology σ(∆, δ) on 2Zequals τ− V∨σ(∆, δ)+. The upper ∆U-topology τ(∆U)+on 2Zhas a base {W+:Wc∈∆ or clW ∈∆}. The upper proximal ∆Utopology σ(∆U, δ)+on 2Zhas a base {W++ :Wc∈∆ or clW ∈∆}. The ∆U-topology τ(∆U) on 2Zequals τ− V∨τ(∆U)+([8] and [16]). The proximal ∆U-topology σ(∆U, δ) on 2Zequals τ− V∨σ(∆U, δ)+. Special cases: (1) Vietoris and proximal topologies: when ∆ = CL(Z), the upper Vietoris topology τ(V)+=τ(CL(Z))+; the Vietoris topology τ(V) = τ(CL(Z)); the upper proximal topology σ(δ)+=σ(CL(Z), δ)+and the proximal topology σ(δ) = σ(CL(Z), δ) = σ, if δis understood. The paper [7] deals with only metric proximities, and [18] remains unpublished. It is not widely known that proximal hypertopologies can be studied in more general situations and not merely in metric spaces. (However, see the recent papers [11], [12] and [16]). We note that the Vietoris topology is itself a proximal topology, i.e. τV= σ(δ0). (2) Fell topology: when ∆ = K(Z), the upper Fell topology (also called the co-compact topology)τ(F)+=τ(K(Z))+; the Fell topology τ(F) = τ(K(Z)); 448 G. Di Maio, E. Meccariello and S. Naimpally the U-topology τ(U) = τ(K(Z)U) (see [8]). When ∆ = K(Z) and δis EF , we have τ(F) = τ(K(Z)) = σ(K(Z), δ) = σ(F, δ) and τ(U) = τ(K(Z)U) = σ(K(Z)U, δ) = σ(U, δ). In this case the Fell topology equals the proximal Fell topology and this explains the reason for several beautiful results. In generalizing results concerning Fell topology to ∆-topologies, we find that some are true in τ(∆) while others are true in σ(∆)! (Also see below about weak topologies). (3) Ball and proximal ball topologies: When Zis a metric space, ∆ = Bis the cobase generated by all finite unions of all closed balls of nonnegative radii and δis the metric proximity, we have the ball topology τ(B) = τ(∆) ([4]); the proximal ball topology σ(B) = σ(∆, δ) ([17]). The proximal ball topology is very close to the Wjisman topology. In fact, the two are equal in metric spaces satisfying some simple conditions that are present in a normed linear space ([17]). For Z=X×Y, when we wish to refer to hypertopologies on 2Y, we use the suffix 2 e.g. τ2(V) denotes the Vietoris topology on 2Y; τ2(F) denotes the Fell topology on 2Y; σ2(δ2) = σ2denotes the proximal topology w.r.t. δ2on 2Yetc.. (4) Weak topologies: If Z=X×Y, then for each of the topologies involving ∆ described above, we also have an associated weak topology wherein ∆ is replaced by ∆1×∆2(see [31]) and we attach the letter ”w”. Thus τ(w∆) = τ(∆1×∆2) and important special examples are: the weak Vietoris topology τ(wV ) = τ(CL(X)×CL(Y)) ⊂τ(V) = τ(CL(Z)); the weak Fell topology τ(wF ) = τ(K(X)×K(Y)) ⊂τ(F) = τ(K(Z)). For single-valued functions with closed graphs, it was shown in [21] that τ(wF) = τ(F). The proof also works for Fand combining this result with the fact that when the proximity δis EF ,τ(F) = σ(F, δ) we have τ(wF) = τ(F) = σ(wF, δ) = σ(F, δ). Graph topologies on closed multifunctions 449 In generalizing McCoy’s results involving Fell topology we find that our generalizations hold if we replace an appropriate member from the above four. (5) Hausdorff-Bourbaki and Attouch-Wets topologies: Definition 1.1. Let Ybe a Tychonoff space, Va compatible uniformity and ∆2⊂CL(Y). (i) For each V∈ V set: VH={(A, B)∈2Y×2Y:A⊂V(B) and B⊂V(A)}. The family {VH:V∈ V} is a base for a uniformity VHon 2Ycalled the Hausdorff-Bourbaki uniformity (cf. [4]) (or the HB-uniformity for short). (ii) Whereas for each D∈∆2and V∈ V set: [D, V ] = {(A, B)∈2Y×2Y:A∩D⊂V(B) and B∩D⊂V(A)}. The family {[D, V ] : D∈∆2and V∈ V} is a base for a filter V∆2 on 2Ycalled the ∆2-Attouch-Wets filter (cf. [5], [6] and [16]) (or the ∆2-AW filter for short). Remark 1.2. Let Ybe a locally compact space, Va compatible uniformity and ∆2=K(Y). Then the corresponding ∆2-AW filter V∆2on 2Yis a uniformity (see [4] or [5]) and it will be denoted with U(F). Moreover, if 2Yis equipped with the Fell topology τ2(F), it is known that U(F) is compatible with τ2(F) (see [4] and [10]). Observe that in this case (2Y, τ2(F)) is a compact Hausdorff space and thus U(F) is the unique uniformity on 2Ycorresponding to the Fell topology and it is generated by all τ2(F)×τ2(F) open neighbourhoods of the diagonal in 2Y×2Y. Thus, if Yis a Tychonoff space with a compatible uniformity V, ∆2⊂ CL(Y) and VHand V∆2the associated HB-uniformity and ∆2-AW filter on 2Yrespectively, then on the space F: (a) A typical basic open set in the HB-uniform convergence topology τ(UC∆1,VH)on ∆1is of the form < f, A, VH>={g∈F: for all x∈A, (f(x), g(x)) ∈VH}, where f∈F,A∈∆1and VH∈ VH. If ∆1=K(X), we get one of the most important topologies, namely the HB-uniform convergence topology on compacta τ(UCC, VH). If we replace Aby X, we get another important topology: the HBuniform convergence topology τ(UC, VH). If ∆1is the family of all finite subsets of X, then we have the pointwise HB-convergence topology τp(VH). When VHis understood, we may omit it and just write τ(UCC) and τ(UC). (b) The topology on Fgenerated by 450 G. Di Maio, E. Meccariello and S. Naimpally {< f, A, M >:f∈F, A ∈∆1and M∈ V∆2} is called the ∆2-AW convergence topology τ(UC∆1,V∆2)on ∆1. If A=X, we have the ∆2-AW convergence topology τ(UC, V∆2). As before, we replace ∆1by Cfor ”compacta”. If ∆1=K(X), we obtain the ∆2-AW convergence topology on compacta τ(UCC, V∆2). If ∆1is the family of all finite subsets of X, then we have the pointwise ∆2-AW convergence topology τp(V∆2). By Remark 1.2 it follows that whenever Yis a locally compact space and 2Yis equipped with the Fell topology τ2(F), then the corresponding ∆2-AW filter U(F) is a uniformity which is independent of the uniformity Vchosen on Y. Note that the topology τ(UCC, U(F)) on Fis just what McCoy calls ”Fell uniform topology (on compact sets)” (see [24]). (6) Pseudo uniform topologies: In [22] and [25] function space topologies akin to uniform topologies were studied. The range space was not necessarily uniformizable. Here we introduce a similar concept. Let Wbe a symmetric neighbourhood of the diagonal in (2Y×2Y, τ2×τ2). For each f∈Fand A∈∆1we set W?(f, A) = {g∈F: for all x∈A, (f(x), g(x)) ∈W}. The topology on Fgenerated by {W?(f, A) : f∈F, A ∈∆1and Wa symmetric τ2×τ2neighbourhood of the diagonal in 2Y×2Y}is the τ2-pseudo uniform topology on ∆1:ps(τ(UC∆1, τ2)). If A=X, we have the pseudo τ2-uniform topology ps(τ(UC, τ2)). As before, if ∆1=K(X), we replace ∆1by Cfor ”compacta” and we have the pseudo τ2-uniform topology on compacta ps(τ(UCC, τ2)). In case (2Y, τ2) is uniformizable and we restrict W’s to symmetric entourages, we do get a uniform topology. This is true as in (5) above or in (6) when Yis a locally compact space and 2Yis equipped with the Fell topology τ2(F) on 2Yand in this case ps(τ(UC∆1, τ2(F)) = τ(UC∆1,U(F)) (cf. above Remark 1.2). Although McCoy got his results in uniform setting, we find that some of his results do not need uniformity at all! Finally, if τ2is a given hypertopology on 2Y, then τp(τ2) is the corresponding τ2-pointwise convergence topology on Fwhich agrees with the pseudo τ2 uniform topology on ∆1ps(τ(UC∆1, τ2)) when ∆1is the family of all finite subsets of X. Graph topologies on closed multifunctions 451 Those interested in more details are referred to [4] for hypertopologies, [30] for proximities, [26] and [28] for function space topologies, [9], [10] and [24] for uniform topologies and convergences on spaces of multifunctions. 2. Basic results. One of the most valuable result in function space topologies is the embedding of the range space in the function space (cf. Theorem 2.1.1, page 15 in [26]). In this section we prove similar results for multifunctions which are of fundamental importance in our work. We need to introduce ”upper” hypertopologies that are specially meant for the family C={X×E:E∈CL(Y)} of constant multifunctions. These topologies depend on ∆2alone, unlike other hypertopologies which depend on either ∆ or ∆1×∆2. We use the suffix r (for range) for such topologies. On CL(Z), we have the upper r-∆2-topology τ(r∆2)+which is generated by the basis {(X×V)+:Vc∈∆2}∪{CL(Z)}. Similarly, we have the upper r-∆2U-topology τ(r∆2U)+which is generated by the basis {(X×V)+:Vc∈∆2or clV ∈∆2}∪{CL(Z)}. If δ2is a LO-proximity on Y, then we define an associated ”proximity” rδ2 on Cby (X×E)(X×V) w.r.t. rδ2iff EVw.r.t. δ2. If δ2is an EF -proximity on Y, then it is easy to see that rδ2on Cis also EF. Naturally we also have the proximal versions: the upper proximal r-∆2-topology σ(r∆2, rδ2)+; the upper proximal r-∆2U-topology σ(r∆2U, rδ2)+. We have on CL(Z) the r-∆2-topology τ(r∆2) = τ(r∆2)+∨τ− V. Similarly the analogues: the r-∆2U-topology τ(r∆2U) = τ(r∆2U)+∨τ− V; the proximal r-∆2-topology σ(r∆2, rδ2) = σ(r∆2, rδ2)+∨τ− Vand the proximal r-∆2U-topology σ(r∆2U, rδ2) = σ(r∆2U, rδ2)+∨τ− V. Let P(Y) and P(Z) denote the set of all subsets of Yand Z, respectively. Consider the map j:P(Y),→ P(Z) defined by j(E) = (X×E)∈ P(Z) . Obviously, (a) j:CL(Y),→ C is a bijection; 452 G. Di Maio, E. Meccariello and S. Naimpally (b) j(V+)⊂[j(V)]+; (c) j(V++ δ2)⊂[j(V)]++ rδ2; (d) j(V−)⊂[j(V)]−; (e) j(E)∈W−and W∈τtogether imply E∈[p2(W)]−, where p2:Z→ Yis the projection. (f) Let ∆1⊂CL(X), D∈∆1,Ua filter on 2Y,Ma symmetric member of U,A∈CL(Y) and f=j(A) = X×A. Then: < f, x, M > ∩C =< f, D, M > ∩C =< f, X, M > ∩C =j(M(A)) for each x∈X. So, we have the following results. Theorem 2.1. Let Xand Ybe Hausdorff spaces with compatible LO-proximities. The following are embeddings: (a) j: (CL(Y), τ2(∆2)+),→(CL(Z), τ(r∆2)+); (b) j: (CL(Y), τ2(∆2U)+),→(CL(Z), τ(r∆2U)+); (c) j: (CL(Y), σ2(∆2, δ2)+),→(CL(Z), σ(r∆2, rδ2)+); (d) j: (CL(Y), σ2(∆2U, δ2)+),→(CL(Z), σ(r∆2U, rδ2)+); (e) j: (CL(Y), τ2(∆2)) ,→(CL(Z), τ(r∆2)); (f) j: (CL(Y), τ2(∆2U)) ,→(CL(Z), τ(r∆2U)); (g) j: (CL(Y), σ2(∆2, δ2)) ,→(CL(Z), σ(r∆2, rδ2)); (h) j: (CL(Y), σ2(∆2U, δ2)) ,→(CL(Z), σ(r∆2U, rδ2)). Remark 2.2. Let Ybe a Tychonoff space, Va compatible uniformity on Y, ∆1⊂CL(X) and ∆2⊂CL(Y). If VHand V∆2are respectively the associated HB-uniformity and ∆2-AW filter on 2Y, then on C: (1) τ(UC, VH) = τ(UC∆1,VH) = τp(VH); (2) τ(UC, V∆2) = τ(UC∆1,V∆2) = τp(V∆2). Thus the following are embeddings: (1a) j: (CL(Y), τ2(VH)) ,→(CL(Z), τ(UC, VH)); (1b) j: (CL(Y), τ2(VH)) ,→(CL(Z), τ(UC∆1,VH)). (2a) j: (CL(Y),V∆2),→(CL(Z), τ(UC, V∆2)); (2b) j: (CL(Y),V∆2),→(CL(Z), τ(UC∆1,V∆2)). Similarly, if Yis a Hausdorff space, τ2a given hypertopology on 2Yand ∆1⊂CL(X), then on C: (3) ps(τ(UC, τ2)) = ps(τ(UC∆1, τ2)) = τp(τ2). Thus the following are embeddings: (3a) j: (CL(Y), τ2),→(CL(Z), ps(τ(UC, τ2))); (3b) j: (CL(Y), τ2),→(CL(Z), ps(τ(UC∆1, τ2))). Lemma 2.3. Let Xand Ybe Hausdorff spaces. Then, on the family Cof constant multifunctions: (a) τ(w∆)+≤τ(r∆2)+≤τ(r∆2U)+≤τ(V)+; and if p2(∆) ⊂∆2, then τ(w∆)+≤τ(∆)+≤τ(r∆2)+≤τ(r∆2U)+≤τ(V)+. Graph topologies on closed multifunctions 453 (b) τ(w∆)+≤τ(∆)+≤τ(∆U)+≤τ(V)+; and if p2(∆) ⊂∆, then τ(w∆)+≤τ(∆)+≤τ(∆U)+≤τ(r∆2U)+≤τ(V)+. (c) τ(w∆) ≤τ(r∆2)≤τ(r∆2U)≤τ(V); and if p2(∆) ⊂∆2, then τ(w∆) ≤τ(∆) ≤τ(r∆2)≤τ(r∆2U)≤τ(V). (d) τ(w∆) ≤τ(∆) ≤τ(∆U)≤τ(V); and if p2(∆) ⊂∆2, then τ(w∆) ≤τ(∆) ≤τ(∆U)≤τ(r∆2U)≤τ(V). Lemma 2.4. Let Xand Ybe Hausdorff spaces with compatible LO-proximities. Then, on the family Cof constant multifunctions: (a) σ(w∆)+≤σ(r∆2)+≤σ(r∆2U)+≤σ+; and if p2(∆) ⊂∆2, then σ(w∆)+≤σ(∆)+≤σ(r∆2)+≤σ(r∆2U)+≤σ+. (b) σ(w∆)+≤σ(∆)+≤σ(∆U)+≤σ+; and if p2(∆) ⊂∆2, then σ(w∆)+≤σ(∆)+≤σ(∆U)+≤σ(r∆2U)+≤σ+. (c) σ(w∆) ≤σ(r∆2)≤σ(r∆2U)≤σ; and if p2(∆) ⊂∆2, then σ(w∆) ≤σ(∆) ≤σ(r∆2)≤σ(r∆2U)≤σ. (d) σ(w∆) ≤σ(∆) ≤σ(∆U)≤σ; and if p2(∆) ⊂∆2, then σ(w∆) ≤σ(∆) ≤σ(∆U)≤σ(r∆2U)≤σ. We say that Zis locally ∆ iff for each z∈Zwith z∈V∈τ, there is D∈∆ with z∈intD ⊂D⊂V. (Note that this is a generalization of local compactness in which case ∆ = K(Z)). Lemma 2.5. Let Xand Ybe Hausdorff spaces with compatible LO-proximities, Z=X×Yand ∆⊂CL(Z)a cover. If Zis locally ∆, then: (a) τ(∆)+=τ(∆U)+if and only if Z∈∆i.e. ∆ = CL(Z). (b) σ(∆)+=σ(∆U)+if and only if Z∈∆i.e. ∆ = CL(Z). Proof. We prove only (a). It suffices to show that τ(∆U)+≤τ(∆)+implies Z∈∆. Suppose Uis a nonempty subset of Zwith A⊂Uand clU ∈∆ (note that such Uexists since Zis locally ∆). Then there is an open subset Vin Z with Vc∈∆ such that A⊂V⊂U. Clearly Z=clU ∪Vc∈∆.  Corollary 2.6. Let Xand Ybe Hausdorff spaces with compatible LO-proximities, Z=X×Yand ∆⊂CL(Z)a cover. If Zis locally ∆, then: (a) τ(∆) = τ(∆U)if and only if Z∈∆i.e. ∆ = CL(Z) (cf. [14], Theorem 3.2). (b) σ(∆) = σ(∆U)if and only if Z∈∆i.e. ∆ = CL(Z). (c) When ∆ = K(Z), we have τ(F) = τ(U)if and only if Xis compact. Remark 2.7. In the following relations, vertical lines show embeddings: 460 G. Di Maio, E. Meccariello and S. Naimpally f(x0), it follows that {fU:U∈ N (x0)}cannot τ(UC∆1,U(σ2(∆2))) converge to f. (c) It follows from the above and the fact that the equality is equivalent to σ2(∆2) = τ2(VH), which in turn, is equivalent to the total boundedness of V. Theorem 3.14. Let Xbe a Hausdorff space, Ya Tychonoff space, Va compatible uniformity on Yand VHthe corresponding HB-uniformity on 2Y. Then on C: (a) If Vis totally bounded, then τ(UC, VH)≤τ(rCL(Y)) ≤τ(V). (b) If Vis not totally bounded, then on C,τ(UC, VH)6≤τ(V). Thus, if on Fτ(UC, VH)≤τ(V), then Vis totally bounded. ([24] Prop. 4.9) If Xis a locally compact space, Ya locally compact completely metrizable space with metric dand Vthe d-metric uniformity, then on Fτ(UCC, VH)≤τ(V)if and only if Xis discrete and Yis compact. Proof. (a) Let U∈ V be open and symmetric and f= (X×E)∈ C. Total boundedness of Vimplies there is a finite set {yk: 1 ≤k≤n})⊂Ysuch that E⊂ n [ k=1 U(yk). Consider a typical τ(UC, VH)- neighbourhood of f, i.e. < f, X, U2>. It is easy to see that [X×U(E)]+∩ n \ k=1 U(yk)−is a τ(wV )- neighbourhood of fwhich is contained in < f, X, U2> . (b) If Vis not totally bounded there is an open U∈ V and a sequence {yk:k∈IN} ⊂ Ysuch that Y6⊂ n [ k=1 U(yk) for each n∈IN. Then it is clear that FC={f=X×E:E⊂Yis finite}is a subset of Cwhich is not dense in (C, τ(UC, VH)) but which is dense in (C, τ(V)).  Proposition 3.15. Let Xbe a Hausdorff space, Ya Tychonoff space, Va compatible uniformity on Yand VHthe corresponding HBuniformity on 2Y. If on Fτ(V)≤τ(UC∆1,VH), then X∈∆1and Yis Atsuji (i.e. ∆1=CL(X) and every real-valued continuous function on Yis uniformly continuous). Proof. First we show X∈∆1. Assume not and let y1,y2be distinct points of Y. Define f=X× {y1}and D=X× {y2}. Then f∈(Dc)+but for any A∈∆1and any V∈ V,< f, A;VH>is not contained in (Dc)+. In fact, choose x0∈(X\A) (which exists since we are assumming X6∈ ∆1) and set g=f∪ {(x0, y2)}then g∈< f, A;VH>, but g6∈ (Dc)+because (x0, y2)∈g(x0)∩D; a contradiction. Then, the result follows from the fact that on Cτ(V)≤τ(UC, VH) if and only if τ2(V)≤τ2(VH) on CL(Y) which in turn is equivalent to Ybeing Atsuji.  Graph topologies on closed multifunctions 461 Next Example suggested by ˇ Lubica Hol´a shows that the converse is not in general true. Example 3.16. Let X= [1,+∞) and Y= [0,1] subspaces of the real line and VHthe Hausdorff metric uniformity on 2Y. Set f=X× {0}and for each natural number ndefine fn=X× { 1 n}. Then the sequence {fn:n∈IN} τ(UC, VH)-converges to fbut it fails to τ(V)-converge to f. In fact, take G={(x, y)∈X×Y:y < 1 x}. Then f∈Gbut fn6∈ G, for each n∈IN. However, if ∆1=K(X) we have the next result. Proposition 3.17. Let Xbe a Hausdorff space, Ya Tychonoff and locally ∆2 space, Va compatible uniformity on Y, then on Fτ(V)≤τ(UCC, VH)if and only if Xis compact and Yis Atsuji. (Cf. [24] Prop. 4.10) If Xis a Hausdorff space, Ya locally compact and completely metrizable space with metric dand Vthe d-metric uniformity, then on Fτ(V)≤τ(UCC, VH)if and only if Xis compact and Yis a topological sum of a compact space and a discrete space. Proof. It is known from [29] that on C(X, Y ), the family of all continuous functions on Xto Y, the graph topology equals the Vietoris topology. Moreover, observe that τ(UCC, VH) on C(X, Y ) equals the compact open topology τk. Thus, from a result analogous to 2(d) page 14 of [26] it follows that on C(X, Y ), τ(V)≤τkif and only if X∈K(X). The statement then follows from the known result that Yis Atsuji if and only if τ2(V)≤τ2(VH).  The proof of the next Proposition is left to the reader. Proposition 3.18. Let Xand Ybe Hausdorff spaces. Then on F τ(∆) = τ(V)if and only if Z∈∆i.e. ∆ = CL(Z). ([24] Prop. 4.11) τ(F) = τ(V)if and only if Zis compact i.e. Xand Yare both compact. To study comparisons between the pseudo uniform topologies with some other topologies we give the following Lemma. Lemma 3.19. Let Xbe a Hausdorff space, Ya Tychonoff and locally ∆2 space, δ1a compatible LO-proximity on X,δ2a compatible EF -proximity on Y,Z=X×Yequipped with the product proximity δ=δ1×δ2,τ2(V−)and σ2(∆2)respectively the lower Vietoris topology and the proximal ∆2-topology on 2Y. Then on F: (a) τp(τ2(V−)) ≤ps(τ(UC∆1, σ2(∆2))) ≤ps(τ(UC, σ2(∆2))). If Yis a Hausdorff and locally ∆2space and 2Yis equipped with the ∆2topology τ2(∆2), then: (b) τp(τ2(V−)) ≤ps(τ(UC∆1, τ2(∆2))) ≤ps(τ(UC, τ2(∆2))). 462 G. Di Maio, E. Meccariello and S. Naimpally Proof. We prove only (a). To show (b) few changes are needed. Suppose < f, {x}, V −>is a τp(τ2(V−)) neighbourhood of f, where V∈τ2 and f∈F. So there is a point y∈f(x)∩V. Since Yis locally ∆2, there is aD∈∆2such that y∈intD ⊂D⊂V. Since the proximity δ2is EF we also have y∈intD ⊂DV. Then W= (V−×V−)∪[(Dc)++ ×(Dc)++] is a symmetric neighbourhood of the diagonal in (2Y×2Y, σ2×σ2). Clearly, f∈W?(f, {x})⊂V−. Theorem 3.20. Let Xbe a Hausdorff space with a compatible LO-proximity δ1,Ya Tychonoff space with a compatible EF-proximity δ2,Z=X×Y equipped with the product proximity δ=δ1×δ2and 2Yequipped with the proximal ∆2-topology σ2(∆2)induced by δ2. If Yis locally ∆2, then on F: (a) σ(w∆, δ)≤ps(τ(UC∆1, σ2(∆2))) ≤ps(τ(UC, σ2(∆2))). If Yis a Hausdorff and locally ∆2space and 2Yis equipped with the ∆2 topology τ2(∆2), then: (b) τ(w∆) ≤ps(τ(UC∆1, τ2(∆2))) ≤ps(τ(UC, τ2(∆2))). Proof. Again it suffices to show (a). By above Lemma and Remark 3.10 it suffices to show that on F σ(w∆, δ)+≤ps(τ(UC∆1, σ2(∆2))) ≤ps(τ(UC, σ2(∆2))). Thus, suppose D=A×Bwhere A∈∆1,B∈∆2and fDc. There is an open set Vin Ywith Bδ2Vand such that fA×V(because δ2is EF ). Set S= (V−×V−)∪[(Bc)++ ×(Bc)++] a symmetric neighbourhood of the diagonal in (2Y×2Y, σ2×σ2). Then f∈S?(f, A)⊂(Dc)++. So the first inclusion follows. The second one is trivial. Clearly (b) follows from above with obvious changes.  Theorem 3.21. Let Xand Ybe Hausdorff spaces with Ylocally ∆2,δ1a compatible LO-proximity on X,δ2a compatible LO-proximity on Yand δ= δ1×δ2the product proximity on Z=X×Y. Let σ2and τ2denote the proximal ∆2-topology and the ∆2-topology on 2Y, respectively. Then on F: (a) If ∆1is the family of all finite subsets of X, then ps(τ(UC∆1, σ2)) ≤σ(w∆) and ps(τ(UC∆1, τ2)) ≤τ(w∆). (b) If ps(τ(UC∆1, σ2)) ≤σ(w∆) or ps(τ(UC∆1, σ2)) ≤τ(w∆), then Xis discrete. Proof. To check (a) observe that if ∆1is the family of all finite subsets of X, then ps(τ(UC∆1, σ2(∆2))) = τp(σ2(∆2)) as well as ps(τ(UC∆1, τ2(∆2))) = τp(τ2(∆2)) and clearly τp(σ2(∆2)) ≤σ(w∆) as well as τp(τ2(∆2)) ≤τ(w∆). (b) We prove only the second part, i.e if ps(UC∆1, τ2(∆2)) ≤τ(w∆), then Xis discrete. Assume not. Then there exists a point x0in Xwich is not isolated. Denote by N(x0) the family of all open neighbourhoods of x0and let y0,y1be two different points in Y. For U∈ N (x0) define fU(x) = {y0, y1}for x6∈ Uand fU(x) = {y0}for x∈U. It is easy to verify that fU∈Fand that the net {fU:U∈ N(x0)}τ(w∆)-converges to fdefined by f(x) = {y0, y1} Graph topologies on closed multifunctions 463 for x∈X. Since {fU(x0) : U∈ N(x0)}does not τ2(V−) converge to f(x0), it follows that {fU:U∈ N (x0)}cannot ps(τ(UC∆1, τ2(∆2))) converge to f. Theorem 3.22. Let Xand Ybe Hausdorff spaces. If Yis locally ∆2, then on Fτ(wV )≤ps(τ(UC∆1, τ2(∆2))) if and only if X∈∆1and Y∈∆2(i.e. ∆1=CL(X)and ∆2=CL(Y)). (Cf. [24] Prop. 4.12) If Xis a Hausdorff space and Yis a locally compact space, then τ(V)≤τ(UCC, U(F)) if and only if Zis compact i.e. Xand Yare both compact. Proof. First observe that from Remark 2.8 (g) it follows that on Cτ(wV )≤ ps(τ(UC∆1, τ2(∆2))) ⇔τ2(V)≤τ2(∆2)⇔Y∈∆2i.e. ∆2=CL(Y). Next, by (b) in Theorem 3.20 to show that τ(wV )≤ps(τ(UC∆1, τ(V))) is equivalent to X∈∆1it suffices to prove that the inequality τ(wV )≤ps(τ(UC∆1, τ2(V)) implies X∈∆1. Assume not. Let y1,y2be distinct points of Y. Set f= X× {y1}and D=X× {y2}. Clearly, D∈CL(X)×CL(Y) and f∈(Dc)+. We claim that for each ps(τ(UC∆1, τ2(V))) neighbourhood W?(f, A) of fthere exists g∈W?(f, A) such that g6∈ (Dc)+. In fact, since A6=Xthere exists some x0∈(X\A). Set g=f∪ {(x0, y2)}. Then g∈W?(f, A) but g6∈ (Dc)+ showing thereby that Xmust be in ∆1. The following Example, due to ˇ Lubica Hol´a, shows that the uniform Hausdorff convergence topology τ(UC, VH) is in general not finer than the pseudo proximal uniform topology ps(τ(UC, σ2)). Example 3.23. In the real line with the usual metric d, set X=[ n∈IN (1 n+ 1,1 n). Let Y=X,Vthe metric uniformity on Yassociated to d,Vn= ( 1 n+ 1,1 n), ηn=1 n−1 n+ 1 and yn∈Vnbe fixed for each n∈IN. Let f:X→Y defined by f(Vn) = yn. Let W=[ n∈IN (V− n×V− n)∪ {∅,∅}. Then Wis a symmetric σ2×σ2open neigbourhood of the diagonal of 2Y×2Ysuch that the corresponding W?(f, X) = {g∈F:∀x∈X(f(x), g(x)) ∈W} 6∈ τ(UC, VH). Assume not, i.e. W?(f, X)∈τ(UC, VH). Then there exists a positive real εsuch that < f, X;ε >⊂W?(f, X), where < f, X;ε >={g∈F: Hd(f(x), g(x)) < ε ∀x∈X}(here Hddenotes the Hausdorff distance associated to d). Let η < ε. For each x∈X, set g(x) = f(x) + ηand let n∈IN be such that ηn< η. Of course g∈< f, X;ε >, but g6∈ W?(f, X). 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