Remarks on SHD spaces and more divergence properties (Q6633947)

A space \(X\) is said to be selectively highly divergent, or SHD for short, if, for every sequence \(\{ U_n : n \in \omega \}\) of nonempty open subsets of \(X\), there is, for each \(n \in \omega\), an \(x_n \in U_n\) such that the sequence \((x_n)\) has no convergent subsequence. The SHD notion was introduced in [\textit{C. D. Jiménez-Flores} et al., ``On selectively highly divergent spaces'', Preprint, \url{arXiv:2307.11992}] and this article continues developing the theory, as well as introduces related properties, and provides examples separating the properties.\N\NThe paper opens with a variety of sufficient conditions that guarantee that \(X^{\kappa}\) is SHD where \(\kappa\) is a cardinal.\N\NTheorem 2 asserts that, for a cardinal \(\kappa \geq \mathfrak s\), where \(\mathfrak s\) is the splitting number, and a dense subspace \(X\) of \(D(2)^\kappa\), the \(\kappa\)-product of the discrete doubleton, the Stone-Čech compactification \(\beta X\) of \(X\) is SHD. From this, is it deduced that the \(\Sigma\)-product \(\Sigma(D(2),p,\kappa)\), where \(p \in D(2)\) and \(\kappa \geq \mathfrak s\), is a non-SHD space such that \(\beta \Sigma(D(2),p,\kappa)\) is SHD.\N\NAfter establishing the above-mentioned result concerning the Stone-Čech compactification, the authors move on to consider relationships concerning the Pixley-Roy hyperspace \(\mathscr F[X]\) of \(X\), expanding on a result of [\textit{A. Bella} and \textit{S. Spadaro}, Appl. Gen. Topol. 25, No. 1, 41--46 (2024; Zbl 1537.54005)] which asserts that \(\mathscr F[X]\) is SHD whenever \(X\) is SHD.\N\NTheorem 5, along similar lines to Theorem 2, asserts that \(\mathscr F[X]\) is SHD whenever \(X\) is a dense subspace of \(\Sigma(D(2),p,\kappa)\), for \(p \in D(2)\) and \(\kappa \geq \mathfrak s\). As a consequence, \(\Sigma(D(2),p,\kappa)\), where \(p \in D(2)\) and \(\kappa \geq \mathfrak s\), is a non-SHD space such that \(\mathscr F[\Sigma(D(2),p,\kappa)]\) is SHD.\N\NThe remainder of the paper dives into an investigation relating to properties naturally inspired by the SHD property. Various relationships involving these properties between \(X\) and \(\mathscr F[X]\) are established. Moreover, relationships with topological groups and, in particular \(C_p(X)\), are detailed.\N\NThe paper ends with a list of open questions.