On two questions on selectively highly divergent spaces

TL;DR

This paper investigates the properties of selectively highly divergent (SHD) spaces, providing answers to four open questions about their structure and behavior in topology.

Contribution

It offers new insights and resolutions to four open questions regarding the nature and characteristics of SHD spaces in topology.

Findings

Resolved four open questions about SHD spaces

Clarified conditions under which sequences in SHD spaces have no convergent subsequences

Enhanced understanding of the structure of selectively highly divergent spaces

Abstract

A topological space $X$ is selectively highly divergent (SHD) if for every sequence of non-empty open subsets $\{U_n: n\in \omega \}$ of $X$, we can pick a point $x_n\in U_n$, for every $n<\omega$, such that the sequence $\{x_n: n\in\omega\} $ has no convergent subsequences. In this note we answer four questions related to this notion asked in ArXiv:2307.11992.