Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

ON TYCHONOFF-TYPE HYPERTOPOLOGIES

GEORGI DIMOV, FRANCO OBERSNEL, AND GINO TIRONI

Abstract. In 1975, M. M. Choban [5] introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov [8] used a generalized version of this new topology, calling it Tychonofftype topology. The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family M of subsets of a set X. When M contains all singletons, a description of all Tychonofftype topologies O on M is given. The continuous maps of a special form between spaces of the type (M, O) are described in an isomorphism theorem. The problem of commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology is investigated as well. Some topological properties of the hyperspaces (M, O) with Tychonoff-type topologies O are briefly discussed.

1. Introduction

In 1975, M. M. Choban [5] introduced a new topology on the set of all closed subsets of a topological space for obtaining a generalization of the famous Kolmogoroff Theorem on operations on sets. This new topology is similar to the Tychonoff topology (known also as upper Vietoris topology, or upper semi-finite topology ([13]), or kappa-topology) but is weaker than it. In 1998, G. Dimov and D. Vakarelov [8] used a generalized version of this new topology for proving an isomorphism theorem for the category of all Tarski

2000 Mathematics Subject Classification. Primary 54B20, 54B05; Secondary 54B30, 54D10, 54G99.

Key words and phrases. Tychonoff topology, Tychonoff-type topology, T-space, commutative space, O-commutative space, M-cover, M-closed family, P_{∞}-space.

The first author was partially supported by a Fellowship for Mathematics of the NATO-CNR Outreach Fellowships Programme 1999, Bando 219.32/16.07.1999.

The second and the third authors were supported by the National Group “Analisi reale” of the Italian Ministry of the University and Scientific Research at the University of Trieste.

Georgi Dimov, Franco Obersnel and Gino Tironi, On Tychonoff-type hypertopologies, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 51–70, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204121.

consequence systems. This generalized version was called Tychonoff-type topology.

The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family M of subsets of a set X. When M is a natural family, i.e. it contains all singletons, a description of all Tychonoff-type topologies O on M is given (see Proposition 2.32). For doing this, the notion of T-space is introduced. The natural morphisms for T-spaces are not enough to describe all continuous maps between spaces of the type (M, O), where M is a natural family and O is a Tychonoff-type topology on it; we obtain a characterization of those continuous maps which correspond to the morphisms between T-spaces. This is done by defining suitable categories and by proving that these categories are isomorphic (see Theorem 2.37). In such a way we extend to any natural family M on X the corresponding result obtained in [8] for the family Fin(X) of all finite subsets of X. We investigate also the problem of commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology, i.e. when the hyperspace of any subspace A of a topological space Y is canonically representable as a subspace of the hyperspace of Y. Such investigations were done previously by H.-J. Schmidt [14] for the lower Vietoris topology, by G. Dimov [7, 6] for the Tychonoff topology and for the Vietoris topology, and by B. Karaivanov [12] for other hypertopologies. We study also such a problem for a fixed subspace A of Y. Some results of [7, 6, 15] are generalized. Finally, we study briefly some topological properties (separation axioms, compactness, weight, density, isolated points, P_{∞}) of the hyperspaces (M, O) with Tychonoff-type topologies O. Some results of [10, 8] are generalized.

Let us fix the notations.

Notations 1.1. We denote by ω the set of all positive natural numbers, by ℝ — the real line, and by ℤ — the set of all integers. We put ℕ = ω ∪ {0}.

Let X be a set. We denote by P(X) the set of all subsets of X. Let M, A ⊆ P(X) and A ⊆ X. We will use the following notations:

• A_{M}^{+} := {M ∈ M : M ⊆ A};

• A_{M}^{+} := {A_{M}^{+} : A ∈ A};
• Fin(X) := {M ⊆ X : |M| < ℵ_{0}};
• Fin_{n}(X) := {M ⊆ X : |M| ≤ n}, where n ∈ ω.

We will denote by A^{∩} (respectively by A^{∪}) the closure under finite intersections (unions) of the family A. In other words,

• A^{∩} := {⋂_{i=1}^{k} A_{i} : k ∈ ω, A_{i} ∈ A} and

• A^{∪} := {⋃_{i=1}^{k} A_{i} : k ∈ ω, A_{i} ∈ A}. Let (X, T ) be a topological space. We put

• CL(X) := {M ⊆ X : M is closed in X, M ≠ ∅} and

• Comp(X) := {M ⊆ X : M is compact}.

The closure of a subset A of X in (X, T ) will be denoted by cl_{X}A or \overline{A}^{X}; as usual, for U ⊆ A ⊆ X, we put

• Ex_{A,X}U := X \ cl_{X}(A \ U).

By a base of (X, T ) we will always mean an open base. The weight (resp., the density) of (X, T ) will be denoted by w(X, T ) (resp., d(X, T )).

If C denotes a category, we write X ∈ |C| if X is an object of the category C.

For all undefined here notions and notations, see [9] and [11].

2. Hypertopologies of Tychonoff-type

Fact 2.1. Let X be a set and M, A ⊆ P(X). Then:

(a) ⋂ A_{M}^{+} = (⋂ A)_{M}^{+} ;

(b) A ⊆ B implies that A_{M}^{+} ⊆ B_{M}^{+} for all A, B ⊆ X.

Definition 2.2. Let (X, T ) be a topological space and let M ⊆ P(X). The topology O_{T} on M, having as a base the family T_{M}^{+}, will be called Tychonoff topology on M generated by (X, T ). When M = CL(X), then O_{T} is just the classical Tychonoff topology on CL(X).

Let X be a set and M ⊆ P(X). A topology O on M is called a Tychonoff topology on M if there exists a topology T on X such that T_{M}^{+} is a base of O.

Definition 2.3. Let X be a set and M ⊆ P(X). A topology O on the set M is called a topology of Tychonoff-type on M if the family O ∩ P(X)_{M}^{+} is a base for O.

Clearly, a Tychonoff topology on M is always a topology of Tychonofftype on M, but not viceversa (see Example 2.42).

Fact 2.4. Let X be a set, M ⊆ P(X) and O be a topology of Tychonofftype on M. Then the family B_{O} := {A ⊆ X : A_{M}^{+} ∈ O} is closed under finite intersections, X ∈ B_{O}, and, hence, B_{O} is a base for a topology T_{O} on X. The family (B_{O})_{M}^{+} is a base of O.

Definition 2.5. Let X be a set, M ⊆ P(X) and O be a topology of Tychonoff-type on M. We will say that the topology T_{O} on X, introduced in Fact 2.4, is induced by the topological space (M, O).

Proposition 2.6. Let X be a set and M ⊆ P(X). A topology O on M is a topology of Tychonoff-type if and only if there exists a topology T on X and a base B for T (which contains X and is closed under finite intersections) such that B_{M}^{+} is a base for O.

Proof. Suppose O is a topology of Tychonoff-type on M. Then the topology T_{O} induced by the topological space (M, O) (see Fact 2.4 and Definition 2.5) and the base B_{O} have the required property.

Conversely, suppose T and B are given as in the statement. Then B_{M}^{+} is a base for O, and therefore also O ∩ P(X)_{M}^{+} is a base for O. □

Definition 2.7. Let X be a set and M, B ⊆ P(X). When B_{M}^{+} is a base for a topology O_{B} on M, we will say that B generates a topology on M. (Obviously, the topology O_{B} is of Tychonoff-type.

Proposition 2.8. Let X be a set, M ⊆ P(X) and B ⊆ P(X). The family B generates a topology O_{B} on M if and only if the family B satisfies the following conditions: (MB1) For any M ∈ M there exists a U ∈ B such that M ⊆ U; (MB2) For any U_{1}, U_{2} ∈ B and any M ∈ M with M ⊆ U_{1} ∩ U_{2} there exists

a U_{3} ∈ B such that M ⊆ U_{3} ⊆ U_{1} ∩ U_{2}.

Proof. It follows from Proposition 1.2.1 [9]. □

Corollary 2.9. Let X be a set and M, B ⊆ P(X). If B = B^{∩} and X ∈ B, then B generates a Tychonoff-type topology on M.

Definition 2.10. Let X be a set and M ⊆ P(X). We say that M is a natural family in X if {x} ∈ M for all x ∈ X.

Corollary 2.11. Let X be a set and M be a natural family in X. If B ⊆ P(X) generates a topology on M (see Definition 2.7), then B is a base for a topology on X.

Proof. By Proposition 2.8, B satisfies the conditions (MB1) and (MB2). Since M is natural, this clearly implies that B satisfies the hypotesis of Proposition 1.2.1 [9]. So B is a base for a topology on X. □

Remark 2.12. Trivial examples show that there exist sets X and (nonnatural) families M,B ⊆ P(X) such that B_{M}^{+} is a base for a topology on M but

(a) ⋃ B ≠ X, so that B cannot serve even as subbase of a topology on X (take X = {0, 1}, M = B = {{0}});

(b) B is not a base of a topology of X, although ⋃ B = X (take X = {0, 1, 2}, M = B = {{0, 1}, {0, 2}}).

The example of (b) shows also that if we substitute in 2.11 naturality of M with the condition “⋃ M = X” then we cannot prove that B is a base of a topology on X; however, it is easy to show that the condition “⋃ M = X” implies that ⋃ B = X, i.e. B can serve as a subbase of a topology on X.

Of course, as it follows from Fact 2.1, if B_{M}^{+} is a base of a topology O on M, then B̃ = B^{∩} ∪ {X} is a base for a topology on X and B̃_{M}^{+} is a base of O.

Corollary 2.13. Let (X, T ) be a topological space and let B ⊆ T be a base of (X, T ), closed under finite unions. Then B generates a topology of Tychonoff-type on Fin(X) and Comp(X).

Proof. It follows easily from Proposition 2.8. □

Proposition 2.14. Let X be a set, M,B_{1},B_{2} ⊆ P(X), and suppose that B_{1} and B_{2} generate, respectively, some topologies (of Tychonoff-type) O_{B_{1}} and O_{B_{2}} on M. Then O_{B_{1}} = O_{B_{2}} if and only if the following conditions are satisfied: (CO1) For any M ∈ M and any U_{1} ∈ B_{1} such that M ⊆ U_{1} there exists

U_{2} ∈ B_{2} with M ⊆ U_{2} ⊆ U_{1}; (CO2) For any M ∈ M and any U_{2} ∈ B_{2} such that M ⊆ U_{2} there exists

U_{1} ∈ B_{1} with M ⊆ U_{1} ⊆ U_{2}.

Proof. It follows from 1.2.B [9]. □

Corollary 2.15. Let (X, T ) be a topological space and B_{1}, B_{2} be bases of (X, T ), closed under finite unions. Then they generate (see Corollary 2.13) equal topologies on Fin(X) and Comp(X). In particular, every topology of Tychonoff-type on Fin(X) or on Comp(X), generated by a base of (X, T ) which is closed under finite unions, coincides with the Tychonoff topology generated by (X, T ) on the corresponding set.

Proof. Check that conditions (CO1) and (CO2) of Proposition 2.14 are satisfied. □

Corollary 2.16. Let (X, T ) be a topological space, M ⊆ P(X), B ⊆ T , and suppose that B generates a topology of Tychonoff-type O on M. Then O is the Tychonoff topology on M generated by (X, T ) if and only if for all M ∈ M and for all V ∈ T such that M ⊆ V, there exists U ∈ B with M ⊆ U ⊆ V. In this case we will say that B is an M-base for (X, T ). Clearly, if M is a natural family, then every M-base of (X, T ) is also a base of (X, T ).

Proof. Put B_{1} := T and B_{2} := B. Then condition (CO2) of Proposition 2.14 is trivially satisfied. The condition required in the statement is exactly condition (CO1). □

Definition 2.17. Let X be a set, M ⊆ P(X) and A ⊆ X. A family U ⊆ P(X) will be called an M-cover of A if A = ⋃ U and for all M ∈ M with M ⊆ A there exists some U ∈ U such that M ⊆ U.

Proposition 2.18. Let X be a set and M, A ⊆ P(X). Then the following conditions are equivalent:

(U1) For all U ∈ A and for all x ∈ U, there exists an M ∈ M with

x ∈ M ⊆ U. (U2) For any U ∈ A∪M and for any subfamily {U_{δ} : δ ∈ Δ} of A∪M, the

equality U_{M}^{+} = ⋃_{δ∈Δ} (U_{δ})_{M}^{+} holds if and only if the family {U_{δ}}_{δ∈Δ} is an M-cover of U.

Proof. Observe that, trivially, in condition (U1) we can replace the requirement ‘for all U ∈ A” with “for all U ∈ A ∪ M”.

(U1)⇒(U2). Let U_{M}^{+} = ⋃_{δ∈Δ} (U_{δ})_{M}^{+}, with U,U_{δ} ∈ A ∪ M for all δ ∈ Δ. We will prove first that ⋃_{δ∈Δ} U_{δ} = U.

Let x ∈ ⋃_{δ∈Δ} U_{δ}. Then there exists a δ ∈ Δ such that x ∈ U_{δ}. By assumption, there exists an M ∈ M with x ∈ M ⊆ U_{δ}. Hence M ∈ (U_{δ})_{M}^{+}. Since U_{M}^{+} = ⋃_{δ∈Δ} (U_{δ})_{M}^{+}, we obtain that M ⊆ U. Thus x ∈ U. Therefore, ⋃_{δ∈Δ} U_{δ} ⊆ U.

Conversely, let x ∈ U. By assumption, there exists an M ∈ M such that x ∈ M ⊆ U. Hence M ∈ U_{M}^{+} = ⋃_{δ∈Δ} (U_{δ})_{M}^{+}. Therefore there exists a δ ∈ Δ such that M ∈ (U_{δ})_{M}^{+}, i.e. M ⊆ U_{δ} and x ∈ U_{δ} ⊆ ⋃_{δ∈Δ} U_{δ}.

We have verified that ⋃_{δ∈Δ} U_{δ} = U.

Suppose M ∈ M and M ⊆ U. Then M ∈ U_{M}^{+} and therefore there exists some γ ∈ Δ with M ∈ (U_{γ})_{M}^{+}. Hence M ⊆ U_{γ}.

This shows that the family {U_{δ}}_{δ∈Δ} is an M-cover of U.

The other implication can be easily proved. (Let’s remark that condition (U1) is not used in the proof of this last implication.)

(U2)⇒(U1). Suppose U ∈ A and x ∈ U. Clearly, we have

U_{M}^{+} = ⋃{M_{M}^{+} : M ∈ M, M ⊆ U}.

Then, by assumption, the family {M ∈ M : M ⊆ U} is an M-cover of U. Therefore U = ⋃{M : M ∈ M, M ⊆ U}. Hence there exists an M ∈ M with x ∈ M ⊆ U. □

Proposition 2.19. Let X be a set and M ⊆ P(X). Then the following conditions are equivalent:

(a) M is a natural family;

(b) For any U ⊆ X and for any subfamily {U_{δ} : δ ∈ Δ} of P(X), the equality U_{M}^{+} = ⋃_{δ∈Δ} (U_{δ})_{M}^{+} holds if and only if the family {U_{δ}}_{δ∈Δ} is an M-cover of U.

Proof. Put A = P(X) in Proposition 2.18. □

Proposition 2.20. Let X be a set, M ⊆ P(X), O be a Tychonoff topology on M generated by a topology T on X and M be a network in the sense of Arhangel′skiĭ for T_{O}. Then T = B_{O} and O is generated by a unique topology on X, namely by T_{O} (see Fact 2.4 for the notation B_{O} and T_{O}).

Proof. We only need to show that B_{O} ⊆ T . Assume A ∈ B_{O}. Then A_{M}^{+} ∈ O. Since T generates O, we have A_{M}^{+} = ⋃_{δ∈Δ} (U_{δ})_{M}^{+}, where U_{δ} ∈ T for all δ ∈ Δ. Clearly U_{δ} ∈ B_{O} for all δ ∈ Δ. By Proposition 2.18, we obtain that A = ⋃_{δ∈Δ} U_{δ} and therefore A ∈ T . □

Remark 2.21. Trivial examples show that there exist sets X, families M ⊆ P(X) and Tychonoff topologies on M which are generated by more than one topology on X.

Corollary 2.22. Let X be a set, M ⊆ P(X), O be a topology of Tychonofftype on M and M be a network in the sense of Arhangel′skiĭ for T_{O}. Then O is a Tychonoff topology on M if and only if B_{O} = T_{O} (see Fact 2.4 for the notations B_{O} and T_{O}).

Proof. Suppose B_{O} = T_{O}. Then the topology O is generated by the topology T_{O} on X and hence, by definition, O is a Tychonoff topology on M.

Suppose O is a Tychonoff topology on M. Then O is generated by some topology T on X. By Proposition 2.20, we get T = B_{O}. Hence T_{O} = B_{O}. □

Corollary 2.23. Let X be a set, M be a natural family in X and O be a topology of Tychonoff-type on M. Then O is a Tychonoff topology on M if and only if B_{O} = T_{O}.

Proof. A natural family M satisfies the hypothesis of Corollary 2.22. □

Proposition 2.24. Let (X, T ) be a topological space, M be a natural family in X, B ⊆ T and suppose that B generates a topology O_{B} on M. Then

B_{O_{B}} = {A ⊆ X : A is M-covered by a subfamily of B},

B_{O_{B}} ⊆ T and B^{∩} ⊆ B_{O_{B}} (see Fact 2.4 for the notation B_{O_{B}}).

Proof. It follows from Proposition 2.19 and Fact 2.4. □

Proposition 2.25. Let (X, T ) be a topological space, M be a natural family in X, B be a base for (X, T ) and suppose that B generates a topology O_{B} on M. Let T_{O_{B}} be the topology on X induced by (M, O_{B}). Then T_{O_{B}} = T .

Proof. We have, by Proposition 2.24, that B ⊆ B_{O_{B}} ⊆ T_{O_{B}}. Thus T ⊆ T_{O_{B}}. As it is shown in Proposition 2.24, B_{O_{B}} ⊆ T and hence T_{O_{B}} ⊆ T . So, T = T_{O_{B}}. □

Example 2.26. Let us show that in Proposition 2.25 the requirement “M is a natural family” is essential.

Let X = (0, 1) ⊂ ℝ be the open unit interval with the usual topology, M = {[a, b] : 0 < a < b < 1}, B = {(a, b) : 0 < a < b < 1}. Then the family B satisfies conditions (MB1) and (MB2). Consider the set A = (\frac{1}{2}, \frac{3}{4}) ∪ \{\frac{1}{4}\}. We have A_{M}^{+} = (\frac{1}{2}, \frac{3}{4})_{M}^{+} ∈ B_{M}^{+} and therefore A ∈ B_{O_{B}} even though A is not open in X.

Definition 2.27. Let X be a set and M, U ⊆ P(X). We will say that U is an M-closed family if for all A ⊆ X such that A is M-covered by some subfamily of U, we have that A ∈ U.

Proposition 2.28. Let X be a set and M, M′, U ⊆ P(X), M ⊆ M′. Suppose that U is an M-closed family. Then U is an M′-closed family too.

Proof. Suppose A ⊆ X is M′-covered by some subfamily U′ of U. Since M ⊆ M′, the set A is also M-covered by U′. By the hypothesis, U is an M-closed family. Hence A ∈ U. □

Proposition 2.29. Let X be a set, M be a natural family in X and O be a topology of Tychonoff-type on M. Then B_{O} is an M-closed family (see Fact 2.4 for the notation B_{O}).

Proof. It follows from Proposition 2.19. □

Proposition 2.30. Let (X, T ) be a topological space, M be a natural family in X and B ⊆ T be an M-closed base of (X, T ). Suppose that B generates a topology O_{B} on M. Then B_{O_{B}} = B, X ∈ B and B^{∩} = B (see Fact 2.4 for the notation B_{O_{B}}).

Proof. Obviously, B_{O_{B}} ⊇ B. Let us show that B_{O_{B}} ⊆ B.

Let A ∈ B_{O_{B}}. Then, by Proposition 2.24, A is M-covered by some subfamily of B and, since B is M-closed, we conclude that A ∈ B. So, B_{O_{B}} = B. Now Fact 2.4 implies that X ∈ B and B^{∩} = B. □

Definition 2.31. Let X be a set and M, B ⊆ P(X). The ordered triple (X, B, M) will be called a T -space if B is an M-closed family, X ∈ B and B^{∩} = B.

Note that if (X, B, M) is a T -space, then B is a base for a topology on X.

Proposition 2.32. Let X be a set and M be a natural family in X. Let T T T (X, M) be the set of all topologies of Tychonoff-type on M. Denote by T−Sp(X, M) the set of all T -spaces of the form (X, B, M). Then there is a bijective correspondence between the sets T T T (X, M) and T−Sp(X, M). Namely, consider the function α : T T T (X, M) → T−Sp(X, M), defined by α(O) = (X, B_{O}, M) (see Fact 2.4 for the notation B_{O}), and the function β : T−Sp(X, M) → T T T (X, M), defined by β ((X, B, M)) = O_{B} (see Corollary 2.9 and Definition 2.7); then α and β are bijections and each one is the inverse of the other one.

Proof. Let us show that the function α is well-defined. Let O ∈ T T T (X, M). By Fact 2.4, the family B_{O} is closed under finite intersections and X ∈ B_{O}. By Proposition 2.29, B_{O} is M-closed. Hence (X, B_{O}, M) ∈ T−Sp(X, M).

We will prove now that the function β is well defined. Let B ⊆ P(X) be an M-closed family, closed under finite intersections and such that X ∈ B. Then, by Corollary 2.9, O_{B} is a topology of Tychonoff-type on M.

Proposition 2.30 gives that B_{O_{B}} = B, i.e. α ∘ β = id_{T−Sp(X,M)}.

To show that β ∘ α = id_{T T T (X,M)}, let O be a topology of Tychonoff-type on M. Then β(α(O)) = O_{B_{O}}. Since O is a topology of Tychonoff-type, (B_{O})_{M}^{+} is a base of O. On the other hand, (B_{O})_{M}^{+} is, by definition, a base of O_{B_{O}}. Hence O = O_{B_{O}}. □

Definition 2.33. We denote by HT (Hypertopologies of Tychonoff-type) the category defined as follows: its objects are all ordered triples (X, M, O) where X is a set, M is a natural family in X and O is a topology of Tychonoff-type on M. To define the morphisms of HT , let (X, M, O), (X′, M′, O′) be objects of HT and f : X → X′ be a function between the sets X and X′. We will say that f generates a morphism f_{H} of HT between (X, M, O) and (X′, M′, O′) if f(M) ⊆ M′ and the induced function on M, f_{m} : (M, O) → (M′, O′), defined by f_{m}(M) := f(M) (where the M on the left-handside is regarded as an element of M and the M on the righthandside is regarded as a subset of X) is continuous. The morphisms of HT are defined to be all f_{H} generated in this way.

Remark 2.34. It is easy to see that not any continuous map between spaces of the type (M, O) appears as some f_{H} (see Definition 2.33 for the notations). Indeed, let (X, T ) be a discrete space having more than one point; then the constant function c : (CL(X), O_{T}) → (CL(X), O_{T}), defined by c(F) = X for all F ∈ CL(X), is continuous but is not of the type f_{H} (here O_{T} is the classical Tychonoff topology on CL(X) (see Definition 2.2)).

Definition 2.35. We denote by T H the category defined as follows: its objects are all T -spaces (X, B, M) (see Definition 2.31). To define the morphisms of T H, let (X, B, M), (X′, B′, M′) be objects of T H and f : X → X′ be a function between the sets X and X′. We will say that f generates a morphism f^{T} of T H between (X, B, M) and (X′, B′, M′) if f(M) ⊆ M′ and f^{−1}(B′) ⊆ B. The morphisms of T H are defined to be all f^{T} generated in this way.

Lemma 2.36. Let X, X′ be sets, M ⊆ P(X) and M′ ⊆ P(X′). Let f : X → X′ be a function such that f(M) ⊆ M′ and let f_{m} : M → M′ be defined by f_{m}(M) := f(M). Then, for all A′ ⊆ X′, we have

f_{m}^{−1} ((A′)_{M′}^{+}) = (f^{−1}(A′))_{M}^{+} .

Proof. Let M ∈ f_{m}^{−1} ((A′)_{M′}^{+}). Then f_{m}(M) ∈ (A′)_{M′}^{+}, i.e. f(M) ⊆ A′. Since M ⊆ f^{−1} (f(M)) ⊆ f^{−1}(A′), we obtain M ∈ (f^{−1}(A′))_{M}^{+}.

Let M ∈ (f^{−1}(A′))_{M}^{+}. Then M ⊆ f^{−1}(A′) and hence f(M) ⊆ A′. Therefore f_{m}(M) ∈ (A′)_{M′}^{+}, i.e. M ∈ f_{m}^{−1} ((A′)_{M′}^{+}). □

Theorem 2.37. The categories HT and T H are isomorphic.

Proof. We define a functor F : HT → T H as follows: for all (X, M, O) ∈ |HT |, we put F ((X, M, O)) := (X, B_{O}, M) (see Fact 2.4 for the notation B_{O}); for all morphisms f_{H} : (X, M, O) → (X′, M′, O′), we put F (f_{H}) := f^{T} (see Definitions 2.33 and 2.35 for the notations f_{H} and f^{T}).

We define a functor G : T H → HT as follows: for all (X, B, M) ∈ |T H|, we put G ((X, B, M)) := (X, M, O_{B}) (see Definition 2.7 for the notation O_{B}); for all morphisms f^{T} : (X, B, M) → (X′, B′, M′), we put G(f^{T}) := f_{H}.

Let us check that F and G are well-defined.

Let O be a topology of Tychonoff-type on M. By Proposition 2.32, the triple (X, B_{O}, M) is a T -space, so that F (X, M, O) ∈ |T H|.

Let f_{H} : (X, M, O) → (X′, M′, O′) be a morphism in HT . Then f(M) ⊆ M′ and the induced function on M, f_{m} : (M, O) → (M′, O′), defined by f_{m}(M) = f(M), is continuous. To check that

f^{T} : F (X, M, O) → F (X′, M′, O′)

(i.e. f^{T} : (X, B_{O}, M) → (X′, B_{O′}, M′)) is a morphism in T H, we need to show that f^{−1}(B_{O′}) ⊆ B_{O}. Let B′ ∈ B_{O′}. Then (B′)_{M′}^{+} ∈ O′. By the continuity of f_{m}, we have that f_{m}^{−1} ((B′)_{M′}^{+}) ∈ O. By Lemma 2.36, f_{m}^{−1} ((B′)_{M′}^{+}) = (f^{−1}(B′))_{M}^{+}. Hence (f^{−1}(B′))_{M}^{+} ∈ O, i.e. f^{−1}(B′) ∈ B_{O}. So, we have proved that F is well-defined. Clearly, F is a functor.

Let now (X, B, M) ∈ |T H|. Then, by Proposition 2.32, the topology O_{B} is a Tychonoff-type topology on M. Hence, G((X, B, M)) ∈ |HT |.

Let f^{T} : (X, B, M) → (X′, B′, M′) be a morphism in T H. Then f(M) ⊆ M′ and f^{−1}(B′) ⊆ B. To check that

f_{H} : (X, M, O_{B}) → (X′, M′, O_{B′})

is a morphism in HT , we need to check that the function f_{m} : (M, O_{B}) → (M′, O_{B′}), defined by f_{m}(M) := f(M), is continuous. By definition (see 2.7), (B′)_{M′}^{+} is a base of the topology O_{B′}. Let B′ ∈ B′. Then (B′)_{M′}^{+} ∈ (B′)_{M′}^{+}. By assumption, f^{−1}(B′) ∈ B. Hence (f^{−1}(B′))_{M}^{+} ∈ O_{B}. Since, by Lemma 2.36, (f^{−1}(B′))_{M}^{+} = f_{m}^{−1} ((B′)_{M′}^{+}), we obtain that f_{m}^{−1} ((B′)_{M′}^{+}) ∈ O_{B}. Therefore, the function f_{m} is continuous. So, G is well-defined. Obviously, G is a functor.

By Proposition 2.32, we have F ∘ G = id_{T H} and G ∘ F = id_{HT} on the objects. The equalities are clearly true for the morphisms. Hence F and G are isomorphisms. □

We recall that a topological space (X, T ) is called a P_{∞}-space (see [1, 8]) if T is closed under arbitrary intersections.

Lemma 2.38. A space (X, T ) is a P_{∞}-space if and only it it has a base B closed under arbitrary intersections.

Proof. Assume T has a base B closed under arbitrary intersections. Let U ⊆ T . Since ∅ is an open set, we can assume that ⋂ U ≠ ∅. Let x ∈ ⋂ U. For any U ∈ U, let B_{U} ∈ B be such that x ∈ B_{U} ⊆ U. Then

x ∈ ⋂{B_{U} : U ∈ U} ⊆ ⋂ U

and, by assumption, ⋂{B_{U} : U ∈ U} ∈ B. Hence, ⋂ U ∈ T . □

Proposition 2.39. Let (X, M, O) ∈ |HT |. The topological space (M, O) is a P_{∞}-space if and only if the family B_{O} is closed under arbitrary intersections. If (M, O) is a P_{∞}-space then (X, T_{O}) is a P_{∞}-space. If O is a Tychonoff topology on M, then (M, O) is a P_{∞}-space if and only if (X, T_{O}) is a P_{∞}-space (see Fact 2.4 for the notations B_{O} and T_{O}).

Proof. The first two assertions follow from Lemma 2.38, Fact 2.1(a) and the definitions of B_{O} and T_{O}. The last assertion follows now from Corollary 2.22.

□

Corollary 2.40. Let HT_{∞} be the full subcategory of HT having as objects all triples (X, M, O) ∈ |HT | for which the space (M, O) is a P_{∞}-space. Let T H_{∞} be the full subcategory of T H whose objects are all (X, B, M) ∈ |T H| such that the family B is closed under arbitrary intersections. Then HT_{∞} and T H_{∞} are isomorphic.

Proof. It follows from (the proof of) Theorem 2.37 and Proposition 2.39. □

Example 2.41. We will show that there exists (X, M, O) ∈ |HT | such that (X, T_{O}) is a P_{∞}-space but (M, O) is not a P_{∞}-space.

Let X = ω,

B = {{n} : n ∈ ω} ∪ {A ⊆ ω : |ω \ A| < ℵ_{0}} ∪ {∅},

and P(ω) ⊇ M ⊇ Fin_{2}(ω) \ {∅}. The family B is a base of the discrete topology T on ω, it is closed under finite intersections (but not under infinite intersections) and X ∈ B. Let us show that B is M-closed.

Let {B_{δ}}_{δ∈Δ} be a subfamily of B \ {∅} which is an M-cover of a subset B of X. Without loss of generality, we can assume that there exist at least two indices δ_{1} and δ_{2} such that B_{δ_{1}} ≠ B_{δ_{2}}. Then there is at least one δ ∈ Δ such that |ω \ B_{δ}| < ℵ_{0}; otherwise we would have B_{δ_{1}} = {n_{δ_{1}}}, B_{δ_{2}} = {n_{δ_{2}}} and the set F = {n_{δ_{1}}, n_{d_{2}}}, which belongs to M, would be contained in B without being contained in B_{δ} for any δ ∈ Δ. Therefore |ω \ B| < ℵ_{0} and, hence, B ∈ M.

Put O := O_{B}. Then, by Proposition 2.30, B = B_{O} and hence T = T_{O}. Since B is not closed under arbitrary intersections, we obtain, by Proposition 2.39, that (M, O) is not a P_{∞}-space. Clearly, the space (X, T ) is a P_{∞}-space because it is discrete. Observe that O is not a Tychonoff topology since B_{O} ≠ T_{O}.

Example 2.42. Two more examples of Tychonoff-type, non Tychonoff topologies on some families M ⊆ P(X).

Let X be a set with more than two elements. Let M = Fin(X) \ {∅} and let O = {{{x} : x ∈ A} : A ⊆ X} ∪ {M} ∪ {∅}. Then O is a topology on M.

O is a topology of Tychonoff-type since

O ∩ P(X)_{M}^{+} = {{x} : x ∈ X} ∪ {X_{M}^{+}} ∪ {∅}

is a base for O. Clearly (M, O) is a P_{∞}-space.

We have B_{O} = {{x} : x ∈ X}∪{X}∪{∅}, and therefore T_{O} is the discrete topology. By Proposition 2.22, since B_{O} ≠ T_{O}, O is not a Tychonoff topology on M.

Observe that (M, O) is not a T_{0}-space. In fact, the only neighbourhood of any F ∈ M such that |F | ≥ 2 is M, and we are assuming |X| > 2.

We consider now the natural family M′ = P(X) \ {∅} and we define O as above. O is a Tychonoff-type topology on M′ but it is not a Tychonoff topology. Again T_{O} is the discrete topology on X. Hence M′ = CL((X, T_{O})). We observe as before that (M′, O) is not a T_{0}-space. Note also that the family B_{O} is M′′-closed for every natural family M′′ in X which contains all two-points subsets of X and ∅ ∉ M′′.

We will briefly discuss now some topological properties of the hyperspaces (M, O) with Tychonoff-type topologies O.

Fact 2.43. Let X be a set, M ⊆ P(X) and O be a topology of Tychonofftype on M generated by a subfamily B of P(X). Then:

(a) the topological space (M, O) is a T_{0}-space (resp., T_{1}-space) if and only if for any F, G ∈ M with F ≠ G, there exists a B ∈ B such that either F ⊆ B and G ⊈ B, or G ⊆ B and F ⊈ B (resp., F ⊆ B and G ⊈ B).

(b) if for any x ∈ X and for any F ∈ M with x ∉ F, there exists a B ∈ B such that F ⊆ B and x ∉ B, then (M, O) is a T_{0}-space.

Remark 2.44. Let us note that Fact 2.43(b) implies the following assertion, which was mentioned in [5], section 2 (after Lemma 3) (the requirement that X ∈ Ω has to be added there): if (X, T ) is a regular T_{1}-space, M is a family consisting of closed subsets of (X, T ) and B is a base of (X, T ) such that B = B^{∩} and U ∈ B implies that X \ \overline{U} ∈ B, then (M, O_{B}) is a T_{0}-space.

Fact 2.45. Let (X, M, O) ∈ |HT |. Then the correspondence (X, T_{O}) → (M, O), x ↦ {x}, is a homeomorphic embedding. Hence, we have, in particular, that:

(a) w(X, T_{O}) ≤ w(M, O);

(b) if (M, O) is a T_{0}-space then (X, T_{O}) is a T_{0}-space.

Fact 2.46.

(a) Let X be a set, M ⊆ P(X) be a family such that there exist F, G ∈ M with F ⊂ G and F ≠ G, and let O be a topology of Tychonofftype on M. Then (M, O) is not a T_{1}-space.

(b) If (X, M, O) ∈ |HT | then (M, O) is a T_{1}-space if and only if (X, T_{O}) is a T_{1}-space and M = {{x} : x ∈ X}.

Fact 2.47. Let (X, M, O) ∈ |HT |. Then (M, O) is a compact space if and only if any M-cover of X, consisting of elements of B_{O}, has a finite M-subcover.

Proof. It follows from Proposition 2.19. □

Examples 2.48. There are many examples of “very nice” spaces X with non-T_{0}-hyperspaces (M, O_{B}) (see Examples 2.42 and 3.19). As an example of a non-T_{0}-space (X, T ) with a T_{0}-hyperspace (M, O_{T}), consider the twopoints space X = {0, 1}, with T = M = {∅, X}.

There exist non-compact spaces X such that (CL(X), O_{B}) is a compact non-T_{0}-space (e.g., the space (CL(ℝ), O_{B}), described in Example 3.19).

To get an example of a non-compact space X and a natural family M such that (M, O_{B}) is a compact T_{0}-space, consider X := ℝ with its natural topology, M := Fin_{2}(ℝ) ∪ {ℝ} and take B as in Example 3.19.

As an example of a compact space (X, T ) with a non-compact hyperspace (M, O_{T}), consider the unit interval X = [0, 1] with its natural topology and put M = {{x} : x ∈ (0, 1]}.

The next three propositions are generalizations of, respectively, Propositions 1, 2 and 3 of [10], and have proofs similar to those given in [10]. (Let us note that in Proposition 2 of [10] the requirement “∅ ∉ C” has to be added.)

Proposition 2.49. Let (X, M, O) ∈ |HT |, w(M, O) = ℵ_{0}, (X, T_{O}) be a T_{1}-space, B_{O} be closed under countable unions and M contain all infinite countable closed subsets of (X, T_{O}). Then (X, T_{O}) is a compact space.

Proposition 2.50. Let (X, M, O) ∈ |HT | and ∅ ∉ M. Then d(M, O) = d(X, T_{O}).

Proposition 2.51. Let (X, M, O) ∈ |HT | and ∅ ∉ M. Then (M, O) has isolated points if and only if (X, T_{O}) has isolated points.

3. On O-commutative spaces

3.1. Let (X, T ) be a topological space and A ⊆ X. Recall that A is said to be 2-combinatorially embedded in X (see [4]) if the closures in X of any two disjoint closed in A subsets of A are disjoint.

Definition 3.2. Let (X, T ) be a topological space, A ⊆ X and B ⊆ P(X). We will say that A is 2_{B}-combinatorially embedded in X if for any F ∈ CL(A) and for any U ∈ B with F ⊆ U, there exists a V ∈ B such that \overline{F}^{X} ⊆ V and V ∩ A ⊆ U.

Proposition 3.3. Let (X, T ) be a topological space and A ⊆ X. Then A is 2-combinatorially embedded in X if and only if A is 2_{T} -combinatorially embedded in X.

Proof. (⇒) Let H ∈ CL(A), V ∈ T and H ⊆ V. We put U = V ∩ A and F = A \ U. Then F and H are two disjoint closed subsets of A. Hence, by assumption, they have disjoint closures in X, i.e. \overline{F}^{X} ∩ \overline{H}^{X} = ∅. Let W = X \ \overline{F}^{X}. Then W is open in X, W ∩ A = U = V ∩ A and \overline{H}^{X} ⊆ W.

(⇐) Let F and G be two disjoint closed subsets of A. Put V = X \ \overline{G}^{X}. Then V is open in X and F ⊆ V. Hence, by assumption, there exists an open set W such that \overline{F}^{X} ⊆ W and W ∩ A ⊆ V ∩ A. Let U = A \ G. Then Ex_{A,X}U = V. Hence V ∩ A = U and W ⊆ V. We conclude that \overline{F}^{X} ⊆ W ⊆ V = X \ \overline{G}^{X}, i.e. \overline{F}^{X} ∩ \overline{G}^{X} = ∅. □

Remark 3.4. In Example 3.22 below we will show that there exist spaces (X, T ), subspaces A of X and bases B of T such that A is 2_{B}-combinatorially embedded in X but A is not 2-combinatorially embedded in X.

Proposition 3.5. Let (X, T ) be a T_{1}-space, O be a topology of Tychonofftype on CL(X) and A ⊆ X. Put B_{A} = {U ∩ A : U ∈ B_{O}} (see Fact 2.4 for the notation B_{O}). The family B_{A} generates a topology of Tychonoff-type O_{A} on CL(A). The function i_{A,X} : (CL(A), O_{A}) → (CL(X), O), defined by i_{A,X}(F) := \overline{F}^{X}, is inversely continuous (i.e. it is injective and its inverse, defined on i_{A,X} (CL(A)), is a continuous function) if and only if the set A is 2_{B_{O}}-combinatorially embedded in X.

Proof. The family B := B_{O} is closed under finite intersections and X ∈ B (see Fact 2.4). Hence the family B_{A} is closed under finite intersections and A ∈ B_{A}. Therefore, by Corollary 2.9, B_{A} generates a topology of Tychonofftype O_{A} on CL(A).

The function i_{A,X} : (CL(A), O_{A}) → (CL(X), O) is clearly injective. Denote by g its inverse defined on i_{A,X}(CL(A)), i.e. g : i_{A,X}(CL(A)) → CL(A).

(⇒) Let H ∈ CL(A), U ∈ B and H ⊆ U. Then H ∈ (U ∩ A)_{CL(A)}^{+} ∈ O_{A}.

Since g(\overline{H}^{X}) = H, the continuity of g implies that there exists a V ∈ B such that

\overline{H}^{X} ⊆ V and g(V_{CL(X)}^{+} ∩ i_{A,X}(CL(A))) ⊆ (U ∩ A)_{CL(A)}^{+}.

Then V ∩ A ⊆ U ∩ A. Indeed, let x ∈ V ∩ A. Since X is a T_{1}-space, we obtain that

{x} ∈ V_{CL(X)}^{+} ∩ i_{A,X}(CL(A)) and g({x}) = {x}.

Hence x ∈ U ∩ A. So, A is 2_{B}-combinatorially embedded in X.

(⇐) Let F ∈ i_{A,X}(CL(A)) and g(F) = H. Then F = \overline{H}^{X} and H ∈ CL(A). Let U ∈ B_{A} be such that H ⊆ U. Then there exists a V ∈ B with V ∩A = U. Hence H ⊆ V. Since A is 2_{B}-combinatorially embedded in X, there exists a W ∈ B such that F = \overline{H}^{X} ⊆ W and W ∩ A ⊆ V ∩ A = U. Then F ∈ W_{CL(X)}^{+} ∈ O. We will show that

g(W_{CL(X)}^{+} ∩ i_{A,X}(CL(A))) ⊆ U_{CL(A)}^{+}.

Indeed, let K ∈ CL(A), G = \overline{K}^{X} and G ⊆ W. Then g(G) = K and

K = G ∩ A ⊆ W ∩ A ⊆ V ∩ A = U,

i.e. K ∈ U_{CL(A)}^{+}, as we have to show. Hence, g is a continuous function. □

Corollary 3.6 ([6], Theorem 2.1). If in Proposition 3.5 we take O to be the Tychonoff topology on CL(X) generated by (X, T ) then the function i_{A,X} is inversely continuous if and only if A is 2-combinatorially embedded in X.

Proof. It follows from Propositions 3.5, 2.20 and 3.3. □

Corollary 3.7. Let (X, T ) be a T_{2}-space, A ⊆ X and O be a topology of Tychonoff-type on CL(X) generated by a subfamily of T . Let i_{A,X} be inversely continuous (see Proposition 3.5 for the notation i_{A,X}). Assume that the following condition is satisfied:

(*) For any U ∈ T and for all countable F ∈ CL(A) such that |A \ F | ≥

ℵ_{0} and F ⊆ U, there exists a V ∈ B_{O} with F ⊆ V ⊆ U. Then the set A is sequentially closed.

Proof. Put B := B_{O}. Then, by Proposition 2.24, B ⊆ T . Assume that the set A is not sequentially closed. Then there exists a sequence (x_{n})_{n∈ω} in A and a point x ∈ X \ A such that lim_{n→∞} x_{n} = x. Without loss of generality we can assume x_{n} ≠ x_{m} for all n ≠ m.

Let us consider the sets F = {x_{2n} : n ∈ ω} and G = {x_{2n−1} : n ∈ ω}. Put U = X \\overline{G}^{X}. Then F is a countable closed subset of A, |A\F | ≥ ℵ_{0}, F ⊆ U and U ∈ T . By (*), there exists a V ∈ B such that F ⊆ V ⊆ U. Since we are assuming that the function i_{A,X} is inversely continuous, we obtain, by Proposition 3.5, that the set A is 2_{B}-combinatorially embedded in X. Hence there exists a W ∈ B such that \overline{F}^{X} ⊆ W and W ∩ A ⊆ V ∩ A. Then x ∈ W, because x ∈ \overline{F}^{X}. Since W ∈ T and x is a limit point of G, we have G ∩ W ≠ ∅. However this is a contradiction because

W ∩ A ⊆ V ∩ A ⊆ U = X \ \overline{G}^{X},

and hence G ∩ W = ∅. Therefore, A is sequentially closed. □

Remark 3.8. In Example 3.22 below we will show that condition (*) of Corollary 3.7 is essential, i.e., if we omit it, then the set A could fail to be sequentially closed.

Corollary 3.9 ([6], Corollary 2.3). Let (X, T ) be a T_{2}-space, A ⊆ X, O be the Tychonoff topology on CL(X) generated by (X, T ) and i_{A,X} be inversely continuous (see Proposition 3.5 for the notation i_{A,X}). Then the set A is sequentially closed.

Proof. We have, by Proposition 2.20, that B_{O} = T . Then condition (*) of Corollary 3.7 is trivially satisfied. Hence, by Corollary 3.7, A is sequentially closed. □

Corollary 3.10. Let (X, T ) be a sequential T_{2}-space, A ⊆ X and O be a topology of Tychonoff-type on CL(X) generated by a subfamily of T . Assume that condition (*) of Corollary 3.7 is satisfied. Then the following conditions are equivalent (see Proposition 3.5 for the notation i_{A,X}):

(a) i_{A,X} is a homeomorphic embedding;

(b) i_{A,X} is inversely continuous;

(c) A is closed in X.

Proof. It is clear that (a) implies (b). The implication (c)⇒(a) is true for any X, because if A is a closed subset of X then i_{A,X} is the inclusion map. Let us show that (b) implies (c). By Corollary 3.7, A is sequentially closed. Since X is a sequential space, we obtain that the set A is closed. □

Corollary 3.11 ([6], Corollary 2.4). Let (X, T ) be a sequential T_{2}-space, A ⊆ X and O be the Tychonoff topology on CL(X) generated by (X, T ). Then the following conditions are equivalent (see Proposition 3.5 for the notation i_{A,X}):

(a) i_{A,X} is a homeomorphic embedding; (b) i_{A,X} is inversely continuous;

(c) A is closed in X.

Definition 3.12. Let (X, T ) be a topological space and let O be a topology of Tychonoff-type on CL(X). The space (X, T ) is called O-commutative if for any A ⊆ X the function i_{A,X}, defined in Proposition 3.5, is a homeomorphic embedding.

When O is the Tychonoff topology on CL(X) generated by (X, T ), the notion of “O-commutative space” coincides with the notion of “commutative space”, introduced in [7, 6].

Corollary 3.13. Let (X, T ) be a sequential T_{2}-space, O be a topology of Tychonoff-type on CL(X) generated by a subfamily of T and let condition (*) of Corollary 3.7 be satisfied for every subspace A of X. Then X is O-commutative if and only if X is discrete.

Proof. It follows from Corollary 3.10. □

Corollary 3.14 ([6], Corollary 2.5). If X is a sequential T_{2}-space then X is commutative if and only if X is discrete.

Example 3.15. Let us show that there exist spaces X and topologies O of Tychonoff-type on CL(X) that are not Tychonoff topologies and that satisfy all hypothesis of Corollary 3.13.

Let X = D(ℵ_{1}) be the discrete space of cardinality ℵ_{1}. Let

B = {A ⊂ X : |A| ≤ ℵ_{0}} ∪ {X}

and M = CL(X). Then M is a natural family on X, B is M-closed, B^{∩} = B, X ∈ B and B is a base for the discrete topology on X. Let O_{B} be the topology on M generated by B. Then O_{B} is a topology of Tychonoff-type on M, however it is not a Tychonoff topology. In fact, by Proposition 2.30, B = B_{O_{B}}. Hence B_{O_{B}} ≠ T_{O_{B}} and, by Corollary 2.23, O cannot be a Tychonoff topology. Obviously, B = B_{O_{B}} satisfies condition (*) of Corollary 3.7 for any subspace A of X.

Definition 3.16. Let (X, T ) be a topological space and O be a topology of Tychonoff-type on CL(X). The space (X, T ) is called O-HS-space if, for any A ⊆ X, the function i_{A,X}, defined in Proposition 3.5, is continuous.

When O is the Tychonoff topology on CL(X) generated by (X, T ), the notion of “O-HS-space” coincides with the notion of “HS-space”, introduced in [2, 3].

Corollary 3.17. Let (X, T ) be a T_{1}-space and O be a topology of Tychonofftype on CL(X). Then X is an O-commutative space if and only if X is an O-HS-space and every subset A of X is 2_{B_{O}}-combinatorially embedded in X.

Proof. It follows from Proposition 3.5. □

Corollary 3.18 ([6],Corollary 2.2). A T_{1}-space X is commutative if and only if X is an HS-space and every subspace of X is 2-combinatorially embedded in X.

Proof. It follows from Corollary 3.17 and Proposition 3.3. □

Example 3.19. We will describe two Tychonoff-type, non Tychonoff topologies on two different subfamilies of P(ℝ) generated by the family B of all open intervals of ℝ. One of the resulting spaces will be T_{0} and the other one will not.

Let T be the natural topology on X := ℝ. Then the family B of all open intervals in X is a base for T , it is closed under finite intersections and X ∈ B. Put M := CL(X, T ) and M′ := Fin_{2}(X). They are natural families. The family B is both M′-closed and M-closed. Indeed, let U ⊆ X be M′-covered by a subfamily B_{U} of B. Then U ∈ T and for every x, y ∈ U there exists an open interval (α, β) ∈ B_{U} containing the points x and y. Hence U is a connected open set in ℝ, i.e. U ∈ B. Therefore, B is an M′- closed family. Since M′ ⊂ M, we obtain, by Proposition 2.28, that B is an M-closed family as well.

By Corollary 2.9, B generates Tychonoff-type topologies O_{B} on M and O′_{B} on M′. As it follows from Proposition 2.30, B_{O_{B}} = B_{O′_{B}} = B ≠ T . Hence, by Corollary 2.23, O_{B} and O′_{B} are not Tychonoff topologies on M, respectively M′.

It is easy to see that (M′, O′_{B}) is a T_{0}-space. Indeed, let {x, y} and {u, v} be two distinct elements in M′. We can assume x < x + ε < u ≤ v for some ε > 0. Consider the interval B = (x + ε, +∞). Then {u, v} ∈ B_{M′}^{+} but {x, y} ∉ B_{M′}^{+}.

Let’s prove that (M, O_{B}) is not a T_{0}-space. Put F = {2k : k ∈ ℤ} and G = {2k + 1 : k ∈ ℤ}. Then F, G ∈ M and F ≠ G but the only neighbourhood of both F and G in M is X_{M}^{+} = M.

Example 3.20. In the notations of Example 3.19, we will show that (ℝ, T ) is an O_{B}-HS-space.

We are working now with the space (M, O_{B}) from Example 3.19. We will write simply O instead of O_{B}.

Let A ⊆ X. We have to show that the function

i_{A,X} : (CL(A), O_{A}) → (CL(X), O),

where the topology O_{A} on CL(A) is generated by the family

B_{A} = {A ∩ U : U ∈ B},

is continuous (see Proposition 3.5 for the notation i_{A,X}). Let B ∈ B. We will show that i_{A,X}^{−1}(B_{M}^{+}) is an open set. Take an F ∈ i_{A,X}^{−1}(B_{M}^{+}). Then

F ∈ CL(A) and \overline{F}^{X} ⊆ B. There exists an E ∈ B such that

\overline{F}^{X} ⊆ E ⊆ \overline{E}^{X} ⊆ B

(this is clear if F is bounded, since in this case \overline{F}^{X} is compact; if F is unbounded below, but is bounded above, then B = (−∞, β), for some β ∈ ℝ, and we can pick E = (−∞, γ) with sup F < γ < β; similarly if F is unbounded above but not below; if F is unbounded both above and below then we have B = ℝ and we put E := B). Then

F ∈ (E ∩ A)_{CL(A)}^{+} ⊆ i_{A,X}^{−1}(B_{M}^{+}).

Indeed, let G ∈ (E ∩ A)_{CL(A)}^{+}. Then

\overline{G}^{X} ⊆ \overline{E}^{X} ⊆ B,

i.e. i_{A,X}(G) ∈ B_{M}^{+}.

Remark 3.21. Let us note that a similar proof shows that every subspace Y of (ℝ, T ) is an O_{B_{Y}} -HS-space (see Examples 3.19 and 3.20 for the notations).

More generally, let Y be a topological space and D be a base of Y. We will say that Y is D-normal if for every F ∈ CL(Y) and for every D ∈ D such that F ⊆ D there exists an E ∈ D with F ⊆ E ⊆ \overline{E}^{Y} ⊆ D. Now, arguing as in Example 3.20, we can prove that if Y is a D-normal space, D = D^{∩} and Y ∈ D, then Y is an O-HS-space, where O is the Tychonoff-type topology on CL(Y) generated by D. This generalizes the result of M. Sekanina [15] that any normal space is a HS-space.

Example 3.22. In the notations of Examples 3.19 and 3.20, we will show that the function i_{A,ℝ} is a homeomorphic embedding for any open interval A.

We will argue for A = (0, 1); the proof for any other open interval is similar. We know, by Example 3.20, that the function i_{A,X} is continuous. Therefore we only need to prove that i_{A,X} is inversely continuous. By Proposition 3.5, it is enough to show that the set A is 2_{B}-combinatorially embedded in X. So, let H be a closed subset of (0, 1) and let B = (α, β) ∈ B be such that H ⊆ B. We have to find a D ∈ B such that \overline{H}^{X} ⊂ D and D ∩ (0, 1) ⊆ B. Clearly, \overline{H}^{X} ⊆ [0, 1]. If \overline{H}^{X} ⊂ (0, 1), we can take D = B. If 0 ∈ \overline{H}^{X} but 1 ∉ \overline{H}^{X} then α ≤ 0 and we can put D = (−1, β). If 1 ∈ \overline{H}^{X} but 0 ∉ \overline{H}^{X} then β ≥ 1 and we can put D = (α, 2). If 0, 1 ∈ \overline{H}^{X} then α ≤ 0 and β ≥ 1 and we put D = (−1, 2). Therefore, A is 2_{B}-combinatorially embedded in (ℝ, T ).

Note that A is not 2-combinatorially embedded in (ℝ, T ).

Observe that the triple ((ℝ, T ), A, O) satisfies all hypothesis of Corollary 3.7 except for condition (*), but A is not sequentially closed.

Example 3.23. Let Y ⊆ ℝ. We will say, as usual, that a point x ∈ Y is isolated from the right (left) (in Y) if there exists an ε > 0 such that if we put U = (x, x + ε) (U = (x − ε, x)) then U ∩ Y = ∅. Now, in the notations of Examples 3.19 and 3.20, we have: a subspace Y of (ℝ, T ) is O_{B_{Y}} -commutative if and only if every point of Y is either isolated from the right or from the left.

We first show that a space Y that has a point y_{0} which is non-isolated both from the left and from the right cannot be O_{B_{Y}} -commutative. Indeed, put A = Y \ {y_{0}}. We will prove that A is not 2_{B_{Y}} -combinatorially embedded in Y. By Proposition 3.5, this will imply that the function i_{A,Y} is not inversely continuous and hence the space Y will be not O_{B_{Y}} -commutative. Let H = {y_{n} : n ∈ ω} be a decreasing sequence in Y converging to y_{0}. Then H is a closed subset of A and H ⊂ (y_{0}, +∞) ∩ Y. Suppose that there exists B ∈ B such that cl_{Y} H ⊆ B and B ∩ A ⊆ (y_{0}, +∞). Since y_{0} ∈ cl_{Y} H ⊆ B and y_{0} is not isolated from the left, we have that (B ∩ A) \ (y_{0}, +∞) ≠ ∅, which is a contradiction. Hence, A is not 2_{B_{Y}} -combinatorially embedded in Y.

Now we will show that a space Y having only points which are isolated either from the left or from the right is O_{B_{Y}} -commutative. Let A ⊂ Y. We know, by Remark 3.21, that the function i_{A,Y} is continuous. Hence it is enough to show that it is inversely continuous, i.e., according to Proposition 3.5, that A is 2_{B_{Y}} -combinatorially embedded in Y. So, let H ∈ CL(A) and let H ⊆ B ∩ Y for some B = (α, β). We have to find a D ∈ B such that \overline{H}^{Y} ⊂ D ∩ Y and D ∩ A ⊆ B. We have cl_{Y} H ⊆ \overline{B}^{X} = [α, β]. If cl_{Y} H ⊆ B, we can take D = B and we are done. If α ∈ cl_{Y} H and β ∉ cl_{Y} H then α ∈ Y and α is not isolated from the right, being a limit point of H. Hence, by the assumption, α is isolated from the left. Thus there exists a γ < α such that (γ, α) ∩ Y = ∅. Then D = (γ, β) is the required interval. The other two possible cases are treated analogously.

References

[1] P. S. Alexandroff, Diskrete Räume, Rec. Math. [Mat. Sbornik] N.S. 2 (1937), 501–518.

[2] S. Barov, G. Dimov, and S. Nedev, On a theorem of H.-J. Schmidt, C. R. Acad. Bulgare Sci. 46 (1993), no. 3, 9–11. MR 1 261 974

[3] , On a question of M. Paoli and E. Ripoli, Boll. Un. Mat. Ital. A (7) 10 (1996), no. 1, 127–141. MR 97a:54013

[4] Eduard Čech and Josef Novák, On regular and combinatorial imbedding, Časopis Pěst. Mat. Fys. 72 (1947), 7–16. MR 9,98e

[5] M. M. Čoban, Operations over sets, Sibirsk. Mat. Ž. 16 (1975), no. 6, 1332–1351, 1372. MR 54 #7266

[6] Georgi D. Dimov, On the commutability between hyperspaces and subspaces, and on Schmidt’s conjecture, Rend. Istit. Mat. Univ. Trieste 25 (1993), no. 1-2, 175–194 (1994), Proceedings of the Eleventh International Conference of Topology (Trieste, 1993). MR 96d:54011

[7] , Some remarks on the commutability between hyperspaces and subspaces, and on Schmidt’s conjecture, C. R. Acad. Bulgare Sci. 47 (1994), no. 4, 5–8. MR 1 332 595

[8] Georgi D. Dimov and Dimiter Vakarelov, On Scott consequence systems, Fund. Inform. 33 (1998), no. 1, 43–70. MR 2000b:03226

[9] Ryszard Engelking, General topology, PWN—Polish Scientific Publishers, Warsaw, 1977. MR 58 #18316b

[10] Oskar Feichtinger, Hyperspaces with the kappa topology, Topology Proc. 3 (1978), no. 1, 73–78 (1979), Proceedings of the 1978 Topology Conference (Univ. Oklahoma, Norman, Okla., 1978), I. MR 80k:54011

[11] Peter T. Johnstone, Stone spaces, Cambridge University Press, Cambridge, 1982. MR 85f:54002 [12] Borislav Karaivanov, On the commutability between hyperspaces and subspaces, Ques-

tions Answers Gen. Topology 14 (1996), no. 1, 85–102. MR 97a:54014

[13] Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 13,54f

[14] Hans-Jürgen Schmidt, Hyperspaces of quotient and subspaces. I. Hausdorff topological spaces, Math. Nachr. 104 (1981), 271–280. MR 84e:54011a

[15] Milan Sekanina, Topologies on systems of subsets, General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976), Part B, Soc. Czechoslovak Mathematicians and Physicists, Prague, 1977, pp. 420–424. MR 57 #13826

Department of Mathematics, University of Sofia, Blvd. J. Bourchier 5, 1126 Sofia, Bulgaria

   E-mail address: gdimov@fmi.uni-sofia.bg

Department of Mathematical Sciences and Computer Science, University of Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy

   E-mail address: obersnel@mathsun1.univ.trieste.it
   E-mail address: tironi@univ.trieste.it