Tsugunori Nogura

When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies

Abstract For a given topological space X we consider two topologies on the hyperspace F(X) of all closed subsets of X. The Fell topology T F on F(X) is generated by the family {OVK: V is open in X and K ⊆ X is compact} as a subbase, where OVK = {F ϵ F(X): F ∩ V ≠ O and F ∩ K = O}. The topology T F is always compact, regardless of the space X. The Kuratowski topology T K is the smallest topology on F(X) which contains both the lower Vietoris topology T lV , generated by the family {{F ϵ F(X): F \ Φ ≠ O}: Φ ϵ F(X)} as a subbase, and the upper Kuratowski topology T uK , which is the strongest topology on F(X) such that upper KuratowskiPainleve convergence of an arbitrary net of closed subsets of X to some closed set A implies that the same net, considered as a net of points of the topological space (F(X), T uK ) , converges in this space to the point A. [Recall that a net 〈Aλ〉λ ϵ Λ ⊆ F(X) upper Kuratowski-Painleve converges to A if ∩{ ∪{A μ : μ ⩾ λ} : λ ϵ Λ} ⊆ A .] The inclusion T F ⊆ T K holds for an arbitrary space X, while the equation T F = T K is equivalent to consonance of X, the notion recently introduced by Dolecki, Greco and Lechicki. These three authors showed that complete metric spaces are consonant. In our paper we give an example of a metric space with the Baire property which is not consonant. We also demonstrate that consonance is a delicate property by providing an example of two consonant spaces X and Y such that their disjoint union X ⊕ Y is not consonant. In particular, locally consonant spaces need not be consonant.

The product of 〈αi〉-spaces

Abstract The purpose of this paper is to give answers to the following problems posed by A.V. Arhangelskii: Is the product of 〈α i 〉-spaces 〈α i 〉 for i = 1, 2, 3, 4, 5? Is every countably compact sequential space 〈α i 〉? We also give, under CH, a negative answer to the following problem: If a subspace of the product of finitely many strongly Frechet spaces is Lasnev, then is it metrizable?

Vietoris continuous selections and disconnectedness-like properties

Suppose that X is a Hausdorff space such that its Vietoris hyperspace (F(X), τV ) has a continuous selection. Do disconnectedness-like properties of X depend on the variety of continuous selections for (F(X), τV ) and vice versa? In general, the answer is “yes” and, in some particular situations, we were also able to set proper characterizations.

Characterizations of intervals via continuous selections

We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results.

Characterizations of compact ordinal spaces via continuous selections

Abstract We give a characterization of compact ordinal spaces by means of the existence of a continuous selection which has the property that the value of every nonempty closed set is an isolated point of the closed set.

Metrizability of topological groups having weak topologies with respect to good covers

Abstract A cover C of a topological space X is point-countable (point-finite) if every point of X belongs to at most countably many (at most finitely many) elements of C . We say that a space X has the weak topology with respect to a cover C provided that a set F ⊆ X is closed in X if and only if its intersection F ∩ C with every C ϵ C is closed in C . A space X is an α 4 -space if for every point x ϵ X and any countable family { S n : n ϵ N } of sequences converging to x one can find a sequence S converging to x which meets infinitely many S n . The classical Birkhoff-Kakutani theorem says that a Hausdorff topological group is metrizable if (and only if) it is first countable. Quite recently Arhangelskii generalized this theorem by showing that Hausdorff bisequential topological groups are metrizable (recall that first countable spaces are bisequential). In our paper we generalize these results by showing that a Hausdorff topological group is metrizable if it has the weak topology with respect to a point-finite cover consisting of bisequential spaces. In addition we establish the following theorem each item of which also generalizes both Birkhoff-Kakutanis and Archangelskiis results:

Fréchetness of inverse limits and products

Abstract A class C of topological spaces is said to be almost countably productive if ∏ni = 1 Xi ϵ C for every n ϵ N, then ∏∞i = 1 Xn ϵ C . In this paper we shall show that the classes 〈αi〉, 〈αi-FU〉, i = 1, 2, 3, which were introduced by Arhangelskiǐ, are almost countably productive. As a consequence we shall give a positive answer to the following problem posed by Galvin: If Xn is a w-space for every n ϵ N, must Xω be a w-space?

Extreme selections for hyperspaces of topological spaces

Abstract We study properties of Hausdorff spaces X which depend on the variety of continuous selections for their Vietoris hyperspaces F (X) of closed non-empty subsets. Involving extreme selections for F (X) , we characterize several classes of connected-like spaces. In the same way, we also characterize several classes of disconnected-like spaces, for instance all countable scattered metrizable spaces. Further, involving another type of selections for F (X) , we study local properties of X related to orderability. In particular, we characterize some classes of orderable spaces with only one non-isolated point.

Selection problems for hyperspaces

Publisher Summary This chapter discusses selection problems for hyperspaces. For a T 1 -space X , let ℱ( X ) be the set of all nonempty closed subsets of X. ℱ( X ) is endowed with the Vietoris topology TV , and called the Vietoris hyperspace of X. A space X is orderable (or linearly orderable) if the topology of X coincides with the open interval topology on X generated by a linear ordering on X . A space X is sub-orderable (or generalized ordered) if it can be embedded into an orderable space. A space X is weakly orderable if there exists a coarser orderable topology on X . In all these cases, the corresponding linear order on X is called compatible for the topology of X or a compatible order for X . A selection f : ℱ 2 ( X ) → X is usually called a weak selection for X . This chapter elaborates about weak selections and properties that follow from orderability, and it discusses the concepts of topological well-ordering and selections, in detail. A discussion on selections and disconnectedness-like properties is also presented in the chapter.

Sequential order of product spaces

Abstract We study the sequential order of product spaces. In some classes of sequential spaces we show the product theorems for sequential order. We construct under the continuum hypothesis two Frechet spaces whose product is sequential and its sequential order is ω1.