Selections and countable compactness

Abstract

ABSTRACT. The present paper deals with continuous extreme-like selections for the Vietoris hyperspace of countably compact spaces. Several new results and applications are established, along with some known results which are obtained under minimal hypotheses. The paper contains also a number of examples clarifying the role of countable compactness.

Hyperspace selections avoiding points

“…Answering a question of [8] about the role of countable compactness in Theorem 2.11, see also [35,Problem 3.10], the following complementary result was obtained in [82, Theorems 1.5 and 1.6].…”

“…According to Theorem 5.10 and [8,Theorem 6.2], the answer to the above question is "yes" when X is a countably compact regular space. In fact, it was shown in [8,Corollary 6.4] that for a countably compact regular space with V cs [F (X)] = ∅, total disconnectedness of X is equivalent to zero-dimensionality of X. Evidently, by Theorem 2.14, the result remains valid for every countably compact regular spaces with a continuous weak selection because each suborderable totally disconnected space is zero-dimensional, see page 11. This brings the following special case of Question 34.…”

Selections and approaching points in products

“…Since X is regular, each point is the intersection of the closure of countably many neighbourhoods, hence the space is first countable being countably compact. Thus, a(p, X) ≤ ω for every p ∈ X and, by [2,Corollary 5.4], X will be both Tychonoff and suborderable (in particular, pseudocompact). By [5,Theorem 3.4], X will be totally disconnected.…”