An Excursion with Divergence Properties

TL;DR

This paper explores divergence properties in topology, examining their implications for hyperspaces, and establishing conditions under which certain spaces exhibit discrete selectivity, with applications to rings of continuous functions.

Contribution

It introduces a class of subsemigroups related to divergence properties and characterizes discrete selectivity in hyperspaces, linking these properties to continuous function rings.

Findings

Characterization of the closed discrete selection game on hyperspaces

Identification of conditions for discrete selectivity in Pixley-Roy hyperspaces

Equivalence of divergence properties in rings of continuous functions

Abstract

In this note, we compare and contrast various selective divergence properties such as the properties of being discretely selective and selectively highly divergent. We identify and incorporate a class of subsemigroups of the semigroup of strictly increasing maps from the naturals to themselves. We investigate certain implications for hyperspaces of finite subsets and characterize the closed discrete selection game on a space in terms of a particular selection game on the Vietoris hyperspace of finite subsets of that space. We also isolate some sufficient conditions on a space that guarantee that the corresponding Pixley-Roy hyperspace of finite subsets is discretely selective. We end by noting that the properties of being discretely selective and of being selectively highly divergent are equivalent in rings of continuous functions with standard topologies of uniform convergence.