@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 2, 2003 pp. 281–288 A short note on hit–and–miss hyperspaces René Bartsch and Harry Poppe Dedicated to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. Based on some set-theoretical observations, compact- ness results are given for general hit-and-miss hyperspaces. Compact- ness here is sometimes viewed splitting into “κ-Lindelöfness” and “κ- compactness” for cardinals κ. To focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result, [8], of Som Naimpally, to who this article is hearty dedicated. 2000 AMS Classification: 54B20, 54D30, 54F99. Keywords: hit-and-miss topology, compactness, relative completeness, relative com- pact unions, upper Vietoris topology. 1. Introduction. Let (X,τ) be a topological space. By P(X), P0(X), Cl(X) and K(X) respec- tively we denote the power set, the power set without the empty set ∅, the family of all closed subsets and the set of all compact subsets of X. For B ∈ P(X) and A ⊆ P(X) we define B−A := {A ∈ A|A ∩ B 6= ∅} (hit–set) and B+A := {A ∈ A|A ∩ B = ∅} (miss–set). Specializing A := Cl(X), we get the usual symbols B−,B+. By τl,A we denote the topology for A, generated by the subbase of all G−A,G ∈ τ. Now consider ∅ 6= α ⊆ P(X); by τα,A we denote the topology for A which is generated from the subbase of all B+A,B ∈ α and G−A,G ∈ τ. Of course, for every possible α we have τl,A ⊆ τα,A; for α = Cl(X) we get the Vietoris topology and for α = K(X) we get the Fell topology for A. If α = ∆ ⊆ Cl(X), τα,A is called ∆–topology by Beer and Tamaki [2], and was first introduced by Poppe [10]. By F(X) and F0(X) we denote the set of all filters and ultrafilters, respectively, on a set X (a filter is not allowed to contain the empty set ∅); the symbol F(ϕ) (resp. F0(ϕ)) means the set of all filters (resp. ultrafilters) which contain a given filter ϕ; . x is the filter generated by a singleton {x},x ∈ X. The symbol qτ denotes the convergence structure induced by a topology τ, i.e. qτ := {(ϕ,x) ∈ F(X) ×X|ϕ ⊇ . x∩ τ}, so qτ is a relation between filters and points of a set X. If X is a set, τ, A are subsets of P(X), then we call A weakly complementary w.r.t. τ, iff for every subset σ ⊆ τ there exist a subset B ⊆ A, s.t. ⋃ B∈B B = X\ ⋃ S∈σ S. 282 R. Bartsch and H. Poppe Lemma 1.1. Let X be a set, τ, A ⊆ P(X) and K ⊆ X. Then holds⋃ i∈I Gi ⊇ K =⇒ ⋃ i∈I G −A i ⊇ K −A for every collection Gi, i ∈ I,Gi ∈ τ. If A is weakly complementary w.r.t. τ, then for every collection Gi, i ∈ I,Gi ∈ τ the implication ⋃ i∈I Gi ⊇ K ⇐= ⋃ i∈I G −A i ⊇ K −A holds, too. Proof. Let ⋃ i∈I Gi ⊇ K. A ∈ K −A ⇒ A∩K 6= ∅ ⇒ ∅ 6= A∩ ⋃ i∈I Gi ⇒∃i0 ∈ I : A∩Gi0 6= ∅ ⇒ A ∈ G −A i0 ⇒ A ∈ ⋃ i∈I G −A i . Conversely, let A be weakly complementary w.r.t. τ and ⋃ i∈I G −A i ⊇ K −A . Assume⋃ i∈I Gi 6⊇ K. Then X\ ⋃ i∈I Gi ⊇ K\ ⋃ i∈I Gi 6= ∅ holds, so there is an A ∈ A,A ⊆ X \ ⋃ i∈I Gi with A∩K \ ⋃ i∈I Gi 6= ∅. Thus A ∈ K −A , implying A ∈ ⋃ i∈I G −A i . This yields ∃i0 ∈ I : A∩Gi0 6= ∅ in contradiction to the construction of A. � Corollary 1.2. Let X be a set, τ, A ⊆ P(X) and K ⊆ X. Then holds (1.1) ⋃ i∈I Gi ⊇ K ⇐⇒ ⋃ i∈I G −A i ⊇ K −A for every collection Gi, i ∈ I,Gi ∈ τ if and only if A is weakly complementary w.r.t. τ. Proof. We only have to show, that A is weakly complementary w.r.t. τ, if (1.1) holds. Assume, A is not weakly complementary w.r.t. τ. Then there must be a collection {Gi|i ∈ I}⊆ τ, such that ⋃ {A|A ∈ P(X \ ⋃ i∈I Gi) ∩ A} 6⊇ X \ ⋃ i∈I Gi. Now, we chose K := ( X \ ⋃ i∈I Gi ) \ ⋃ {A|A ∈ P(X \ ⋃ i∈I Gi) ∩A} 6= ∅. Then no element of A, which meets K, can be contained in X \ ⋃ i∈I Gi, i.e. every element of K−A meets ⋃ i∈I Gi, too. So, it must meet a Gi0, i0 ∈ I and consequently it is contained in ⋃ i∈I G −A i . But, by construction, the collection {Gi|i ∈ I} doesn’t cover K, so (1.1) would fail. � Obviously, if for every collection {Gi|i ∈ I} ⊆ τ the complement X \ ⋃ i∈I Gi itself belongs to A, or if all singletons {x},x ∈ X are elements of A, then A is weakly complementary w.r.t. τ. So, if τ is a topology on X, Cl(X) and K(X) are weakly complementary w.r.t. τ. Corollary 1.3. Let (X,τ) be a topological space, K ⊆ X and ∀i ∈ I : Gi ∈ τ. Then holds ⋃ i∈I Gi ⊇ K ⇐⇒ ⋃ i∈I G−i ⊇ K − We have yet another easy, but useful set-theoretical lemma: Lemma 1.4. Let X be a set, A ⊆ P(X) and ϕ ∈ F(X). Assume, A is closed under finite unions of its elements. Then holds ϕ∩ A 6= ∅ ⇐⇒∀ψ ∈ F0(ϕ) : ψ ∩ A 6= ∅ , i.e. a filter contains an A–set, iff each refining ultrafilter contains an A–set. A short note on hit-and-miss hyperspaces 283 Proof. Suppose ∀ψ ∈ F0(ϕ) : ∃Aψ ∈ A : Aψ ∈ ψ. Now, assume ϕ∩ A = ∅. From this automatically follows X 6∈ A. Consider B := {X \A| A ∈ A}. Because of the closedness of A under finite unions, B is closed under finite intersection of its elements, and ∅ 6∈ B, because X 6∈ A. For any F ∈ ϕ,B ∈ B we have F ∩ B 6= ∅, because F ∩ B = ∅ would imply F ⊆ X \ B ∈ A and therefore ϕ ∩ A 6= ∅. So, ϕ ∪ B is a subbase of a filter and consequently, there exists an ultrafilter ψ, containing ϕ∪B, therefore containing ϕ and the complement of every A–set - in contradiction to ∀ψ ∈ F0(ϕ) : ψ ∩ A 6= ∅. The other direction of the statement of the lemma is obvious. � Definition 1.5. Let κ be a cardinal. Then a topological space (X,τ) is called κ-compact, iff every open cover of X with cardinality at most κ admits a finite subcover. (X,τ) is called κ-Lindelöf, iff every open cover of X admits a subcover of cardinality at most κ. A filter is called κ-generated, iff it has a base of cardinality at most κ. A filter ϕ is called κ-completable, iff every subset B ⊆ ϕ with card(B) at most κ fulfills⋂ B∈B B 6= ∅. It is called κ-complete, iff ⋂ B∈B B ∈ ϕ holds under this condition. Proposition 1.6. A topological space (X,τ) is κ-compact, if and only if every κ-generated filter on X has a convergent refining ultrafilter. Proof. Let (X,τ) be κ-compact and ϕ a filter on X with a base B of cardinality at most κ. Assume, all refining ultrafilters of ϕ would fail to converge in X. Then for each element x ∈ X, all refining ultrafilters of ϕ contain the complement of an open neighbourhood of x. But the set of complements of open neighbourhoods of a point x is closed w.r.t. finite unions, thus by Lemma 1.4, ϕ contains the complement of an open neighbourhood of x. So, for each x ∈ X there must exist Ox ∈ τ ∩ . x and Bx ∈ B, s.t. Bx ⊆ X \Ox, implying Bx ⊆ X \Ox and thus X \Bx ⊇ Ox. Now, for each B ∈ B we define OB := X \ B and find, that {OB| B ∈ B} is an open cover of X, because of the preceeding facts. So, there must exist a finite subcover OB1 ∪ ·· · ∪ OBn = X, implying ⋃n i=1(X \ Bi) = X, just meaning ⋂n i=1 Bi = ∅, which is impossible, because all Bi belong to the filter ϕ. So, the assumption must be false; there must exist convergent refining ultrafilters of ϕ. Otherwise, let all κ-generated filters on X have a convergent refining ultrafilter. As- sume, there would exist an open cover C := {Oi ∈ τ| i ∈ I}, ⋃ i∈I Oi = X,card(I) ≤ κ such that all finite subcollections fail to cover X (implying κ to be infinite). But the set of all finite subcollections of the infinite collection C of cardinality at most κ has cardinality at most κ, too. So, B := {X \ ⋃n k=1 Oik| n ∈ IN,ik ∈ I} is a filterbasis of cardinality at most κ, thus there must exist an ultrafilter ϕ ⊇ B, which converges in X - leading to the usual contradiction, because every x ∈ X is contained in an open Ox ∈ C and X \Ox belongs to B ⊆ ϕ. � Analogously we get a characterization of κ-Lindelöf-spaces. Proposition 1.7. If (X,τ) is κ-Lindelöf, then every κ-completable filter on X has a convergent refining ultrafilter. If κ is an infinite cardinal and every κ-complete filter on a topological space (X,τ) has a convergent refining ultrafilter, then (X,τ) is κ-Lindelöf. 284 R. Bartsch and H. Poppe Of course, every κ-complete filter is κ-completable, so we may say, that a topo- logical space (X,τ) is κ-Lindelöf, if and only if each κ-complete filter on X has a convergent refinement. 2. Compactness Properties for Hyperspaces. Lemma 2.1. Let κ be a cardinal, (X,τ) a topological space and let A ⊆ P(X) be weakly complementary w.r.t. τ. If A0 := A \{∅} is κ-Lindelöf (resp. κ-compact) in τl,A0 , then (X,τ) is κ-Lindelöf (resp. κ-compact). Proof. If A is weakly complementary w.r.t. τ, then A0 is, too. So, Corollary 1.2 is applicable. Let {Gi|i ∈ I} be an open cover (resp. an open cover with cardinality at most κ) of X. By Corollary 1.2, then {G−A0i |i ∈ I} is an open cover of X −A0 = A0 (resp. of card. at most κ), so there exists a subset J ⊆ I of cardinality at most κ (resp. a finite subset J), s.t. ⋃ j∈J G −A0 j ⊇ A0 = X −A0 , implying ⋃ j∈J Gj ⊇ X by Corollary 1.2. � Of course, the assumed topology τl,A0 is not really hit-and-miss, because the miss-sets are missed. But every proper hit-and-miss topology would be stronger and therefore it would enforce the desired properties for (X,τ) as well. Lemma 2.2. Let (X,τ) be a κ-compact (resp. κ-Lindelöf ) topological space and assume Cl(X) ⊆ A ⊆ P(X). Then A0 := A \{∅} is κ-compact (resp. κ-Lindelöf ) in τl,A0 . Proof. Let ϕ̂ be a κ-generated (resp. κ-complete) filter on A0. Then, for an arbitrary h ∈A := {g ∈ XP0(X)| ∀M ∈ P0(X) : g(M) ∈ M} the image h(ϕ̂) is a κ-generated (resp. κ-complete) filter on X and consequently it has a τ-convergent refining ultrafilter ψh. Furthermore, there must exist an ultrafilter ψ̂ ⊇ ϕ̂, s.t. h(ψ̂) = ψh. So, the set A := {a ∈ X| ∃f ∈A : (f(ψ̂),a) ∈ qτ} is not empty and consequently the closure A belongs to A0. Now, for any O ∈ τ with A ∈ O−A0 (⇔ A ∩ O 6= ∅) we get A ∩ O 6= ∅ (because of the closure- properties). Now, the assumption O−A0 6∈ ψ̂ would imply O+A0 ∈ ψ̂, yielding ∀f ∈A : X \O ∈ f(ψ̂), thus ∀f ∈A : ∀b ∈ A∩O : (f(ψ̂),b) 6∈ qτ - in contradiction to the construction of A. Thus, O ∈ τ,A ∈ O−A0 always imply O−A0 ∈ ψ̂ and consequently ψ̂ τl,A0 -converges to A. � Definition 2.3. Let (X,τ) be a topological space. A subset A ⊆ X is called weakly relatively complete in X, iff ∀ϕ ∈ F(A) ∩q−1τ (X) : F(ϕ) ∩q −1 τ (A) 6= ∅ , i.e. every filter ϕ on A, which converges in X, has a refinement, converging in A. Proposition 2.4. Let (X,τ) be a topological space and A ⊆ X. Then holds: (a) A is weakly relatively complete in X, iff F0(A)∩q−1τ (X) = F0(A)∩q−1τ (A), i.e. every ultrafilter on A, which converges in X, converges in A. (b) If A is closed in X, then A is weakly relatively complete in X. (c) If A is compact, then A is weakly relatively complete in X. A short note on hit-and-miss hyperspaces 285 (d) If (X,τ) is compact and A is weakly relatively complete in X, then A is compact. (e) If (X,τ) is Hausdorff, then every weakly relatively complete subset A ⊆ X is closed in (X,τ). (f) A is compact iff A is weakly relatively complete and relatively compact. (g) If (X,τ) is κ-compact and A is weakly relatively complete in (X,τ), then A is κ-compact. (h) If (X,τ) is κ-Lindelöf and A is weakly relatively complete in (X,τ), then A is κ-Lindelöf. (i) Weak relative completeness is transitive, i.e. for all A ⊆ B ⊆ X with B weakly relatively complete in (X,τ) and A weakly relatively complete in (B,τ|B), the subset A is weakly relatively complete in (X,τ). There is also a useful description by coverings for weak relative completeness. Lemma 2.5. Let (X,τ) be a topological space and A ⊆ X. Then the following are equivalent: (1) A is weakly relatively complete in X. (2) For every open cover A of A and every element x of X, there is an open neighbourhood Ux,A of x, s.t. Ux,A ∩A is covered by finitely many members of A. (3) For every open cover A of A there exists an open cover A′ ⊇ A of X, such that the intersection of every member of A′ with A can be covered by finitely many members of A, i.e. ∀O ∈ A′ : ∃n ∈ IN,P1, ...,Pn ∈ A : ⋃n i=1 Pi ⊇ O ∩A holds. Proof. (1)⇒(2): Let A ⊆ τ with ⋃ P∈A P ⊇ A be given. For every x ∈ A we can chose a single member of A as open neighbourhood, whose intersection with A is covered by itself. So, assume (2.2) ∃x ∈ X \A : ∀Ux ∈ U(x) ∩ τ : ∀n ∈ IN,P1, ...,Pn ∈ A : Ux ∩A 6⊆ n⋃ i=1 Pi Then B := {(U∩A)\ ⋃n i=1 Pi| U ∈ U(x)∩τ,n ∈ IN,Pi ∈ A} would be closed under finite intersections and thus there would exist an ultrafilter ϕ on A with ϕ ⊇ B. By construction ϕ → x must hold for this ultrafilter, and now by the weak relative completeness of A it follows ∃a ∈ A : U(a) ⊆ ϕ. But A is an open cover of A, so there is an open set P ∈ A with a ∈ P, implying P ∈ ϕ – in contradiction to the construction of ϕ. Thus (2.2) is false and we have ∀x ∈ X \A : ∃Ux ∈ U(x) ∩ τ : ∃n ∈ IN,P1, ...,Pn ∈ A : Ux ∩A ⊆ n⋃ i=1 Pi (2)⇒(3): Note, that (3) is fulfilled with A′ := {Ux| x ∈ X \A}∪ A. (3)⇒(1): For a given ultrafilter ϕ on A with ϕ → x ∈ X assume ϕ 6∈ q−1τ (A). Then ∀a ∈ A : ∃Ua ∈ U(a) ∩ τ : Uca = X \ Ua ∈ ϕ. With these neighbourhoods define A := {Ua| a ∈ A}, which is an open cover of A. By (2) there is an open cover A′ ⊇ A of X such that ∀O ∈ A′ : ∃n ∈ IN,P1, ...,Pn ∈ A : ⋃n i=1 Pi ⊇ O ∩A holds. Now, ϕ → x implies ∃O ∈ A′ : O ∈ ϕ (especially A∩O 6= ∅ follows), and then we have ∃n ∈ IN,P1, ...,Pn ∈ A : O ∩ A ⊆ ⋃n i=1 Pi, implying ∃j ∈ {1, ...,n} : Pj ∈ ϕ 286 R. Bartsch and H. Poppe – in contradiction to the construction of A. So, the assumption ϕ 6∈ q−1τ (A) must be false, showing, that every ultrafilter on A, which converges in X, converges in A. � Theorem 2.6. Let (X,τ) be a topological space, and let α ⊆ P(X) consist of weakly relatively complete subsets of X. Then holds for any A with Cl(X) ⊆ A ⊆ P(X): (A0,τα) is compact ⇐⇒ (X,τ) is compact. Proof. According to Lemma 2.1 we need only to show that (A0,τα) is compact, if (X,τ) is compact. So, assuming (X,τ) to be compact, by Proposition 2.4 every weakly relatively complete subset of X is compact, and we have α ⊆ K(X). Now we will use Alexander’s Lemma: let U be a cover of A0, consisting of subbase elements K +A0 i ,G −A0 j with Ki compact and Gj open. A := X \ ( ⋃ {G|G−A0 ∈ U}) is closed. By construction, A 6∈ G−A0 for any G−A0 ∈ U, so for A 6= ∅ there must exist some K +A0 0 ∈ U with A ∈ K +A0 0 , yielding that K0 ⊆ ⋃ {G|G−A0 ∈ U}; K0 compact ⇒ ∃G1, ...,Gn ∈ U with K0 ⊆ ⋃n k=1 Gk, but then {K +A0 0 }∪{G −A0 1 , ...,G −A0 n } is a cover of A0. If A = ∅, then ⋃ {Gi|G −A0 i ∈ U} = X, so from the compactness of X the existence of some G −A0 1 , ...,G −A0 n ∈ U with X = ⋃n k=1 Gk follows. By Lemma 1.1 then⋃n k=1 G −A0 k = A0 holds. � Many known theorems of compactness w.r.t. the Fell– or the Vietoris–topology follow immediately from the above result. Lemma 2.7. Let (X,τ) be a topological space, A ⊆ P0(X) with Cl(X) ⊆ A and α ⊆ Cl(X). If R ⊆ X is relatively compact in X, then P0(R) ∩ A is relatively compact in (A,τα). Proof. Let B := {O−Ai | i ∈ I,Oi ∈ τ}∪{C +A j | j ∈ J,Cj ∈ α} be an open cover of A by subbase elements of τα. Let O := ⋃ i∈I Oi. If O = X, then there exists finitely many i1, ..., in ∈ I with ⋃n k=1 Oik ⊇ R, because R is relatively compact, and thus ⋃n k=1 O −A ik ⊇ R−A ⊇ P0(R) ∩ A, by Lemma 1.1. If O 6= X, then X \O is nonempty and closed, but not covered by the O−Ai from B. Thus, there must exist a j0 ∈ J with X\O ∈ C+Aj0 , implying Cj0 ⊆ O. Now, we have P0(R)∩A = (P0(R)∩C+Aj0 )∪(P0(R)∩C −A j0 ), and, of course, P0(R)∩C+Aj0 is covered just by C+Aj0 ∈ B. So, we have to find a finite subcover for (P0(R)∩C −A j0 ), if this is not empty. Observe, that R∩Cj0 is relatively compact in X, because it is a subset of R. Furthermore, {Oi| i ∈ I}∪{X\Cj0} is an open cover of X. Thus we find again finitely many i1, .., in ∈ I, s.t. ⋃n k=1 Oik ⊇ R∩Cj0 (because X\Cj0 can be removed from any cover of R∩Cj0 without loosing the covering property). Therefore⋃n k=1 O −A ik ⊇ (R∩Cj0 )−A , by Lemma 1.1. But P0(R)∩C −A j0 ⊆ (R∩Cj0 )−A holds, because any subset of R, which hits Cj0 , automatically hits R∩Cj0 . � 3. Compact Unions. As an interesting application of a simple set-theoretical property, concerning the +-operator, we want to take a brief look at the naturally arising question, whether a union of compact sets itself is compact. Michael showed in [6] that a union of A short note on hit-and-miss hyperspaces 287 closed sets is compact, if the unifying family is compact w.r.t. the Vietoris-topology. Now, the Vietoris-topology is induced by the upper-Vietoris τ+V (miss sets: A +A with Ac ∈ τ) and τl, but τl is not sufficient to enforce compactness of a union of compact sets, as the following example shows: Let X := IR, endowed with euclidian topology, M := {[−m,m]| m ∈ IN}. Then ⋃ M∈M M = IR, is obviously not compact. But every cover of M with elements of the defining subbase for τl must especially cover the element {0} = [0, 0] of M, so it must contain a set O− with 0 ∈ O. Now, every element of M contains the point 0, thus M ⊆ O− follows. So, M is compact in τl by Alexander’s subbase Lemma. And unifying compact sets, τl is not necessary, too, as we will see. Proposition 3.1. Let X be a set, X ⊆ P(X) and M ⊆ X. Then holds⋃ i∈I C +X i ⊇ M =⇒ ⋃ i∈I Cci ⊇ ⋃ M∈M M for every collection Ci, i ∈ I. Proof. For every M ∈ M there must exist an iM ∈ I with M ∈ C +x iM , because of⋃ i∈I C +x i ⊇ M. Thus M ⊆ C c iM ⊆ ⋃ i∈I C c i . � In [5] it was shown Lemma 3.2. Let (X,τ) be a topological space and M ⊆ K(X) compact w.r.t. the upper–Vietoris topology. Then K := ⋃ M∈M M is compact w.r.t. τ. Applying our simple set-theoretical statement, we get a similar result for unions of relatively compact subsets. Lemma 3.3. Let (X,τ) be a topological space, let X be the family of all relatively compact subsets of X and let M ⊆ X be relatively compact in X w.r.t. the upper Vietoris topology. Then R := ⋃ M∈M M is relatively compact in (X,τ). Proof. Let ⋃ i∈I Oi ⊇ X with Oi ∈ τ,i ∈ I an open cover of X. Because of the relative compactness of all P ∈ X, there is a finite subcover Oi1 P , ...,OinP P for ev- ery P ∈ X, i.e. OP := ⋃nP k=1 OikP ⊇ M. Of course, OP ∈ τ and so (OP )c is closed w.r.t. τ. Furthermore, P ∩OcP = ∅, implying P ∈ (O c P ) +X . Thus we have X ⊆ ⋃ P∈X(O c P ) +X , where the (OcP ) +X are open w.r.t. the upper–Vietoris topology. Because of the relative compactness of X w.r.t. the upper–Vietoris topology, there must exist finitely many P1, ...,Pn ∈ X with M ⊆ ⋃n j=1(O c Pj )+X . Now, from Propo- sition 3.1 we get R = ⋃ M∈M M ⊆ ⋃n j=1 OPj , where every OPj is a finite union of members of the original cover {Oi|i ∈ I} by construction. � 288 R. Bartsch and H. Poppe Corollary 3.4. Let (X,τ) be a topological space and let M ⊆ P0(X) consist of relatively compact subsets of X. If M is compact w.r.t. the upper–Vietoris topology, then R := ⋃ M∈M M is relatively compact in (X,τ). Proof. M is compact and therefore relatively compact in every set, which contains M, especially in the family of all relatively compact subsets of X. So, Lemma 3.3 applies. � References [1] Bartsch, R., Dencker, P., Poppe, H., Ascoli–Arzelà–Theory based on continuous convergence in an (almost) non–Hausdorff setting, in ”Categorical Topology”; Dordrecht (1996). [2] Beer, G., Tamaki, R.,On hit-and-miss hyperspace topologies, Commentat. Math. Univ. Carol. 34 (1993), No.4, 717-728. [3] Beer, G., Tamaki, R., The infimal value functional and the uniformization of hit–and–miss hy- perspace topologies, Proc. Am. Math. Soc. 122, No.2 (1994), 601-612. [4] Comfort, W.W., Negrepontis, S., The Theory of Ultrafilters, Berlin (1974). [5] Klein, E., Thompson, A.C., Theory of correspondences. Including applications to mathematical economics. Canadian Mathematical Society Series of Monographs and Advanced Texts (1984). [6] Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. [7] Naimpally, S., Hyperspaces and Function Spaces, Q & A in General Topology 9 (1991), 33-60. [8] Naimpally, S., All Hypertopologies are Hit-and-Miss, Appl. Gen. Topol. 3, No.1 (2001), 45-53. [9] Poppe, H., Eine Bemerkung über Trennungsaxiome in Räumen von abgeschlossenen Teilmengen topologischer Räume, Arch.Math. 16 (1965), 197–199. [10] Poppe, H., Einige Bemerkungen über den Raum der abgeschlossenen Mengen, Fund.Math. 59 (1966), 159–169. [11] Poppe, H., Compactness in General Function Spaces, Berlin (1974). Received November 2001 Revised September 2002 René Bartsch Dept. of Computer Science, Rostock University, Albert-Einstein-Straße 21, 18059 Rostock, Germany E-mail address : rene.bartsch@informatik.uni-rostock.de Harry Poppe Dept. of Mathematics, Rostock University, Universitätsplatz 1, 18055 Rostock, Ger- many E-mail address : harry.poppe@mathematik.uni-rostock.de