When is a Pixley-Roy hyperspace \(SS^{+}\)?
For a space \(X\), let \(PR(X)\) be the space of all nonempty finite subsets of \(X\) with the Pixley-Roy topology, that is, the topology generated by the sets of the form \([A, U]=\{B \in PR(X) : A \subseteq B \subseteq U\}\), where \(A\in PR(X)\) and \(U\) is open in \(X\). This hyperspace was introduced by \textit{C. Pixley} and \textit{P. Roy} [Proc.\ Auburn Topology Conf.\ 1969, 75--85 (1969; Zbl 0259.54022)] to construct a nonseparable Moore space with the countable chain condition. In [Topology Appl. 159, No. 6, 1591--1598 (2012; Zbl 1251.54015)], \textit{M. Sakai} proved that \(PR(X)\) is selectively separable if and only if \(X\) is countable and every finite power of \(X\) has countable fan tightness for finite sets. On the other hand, \textit{D. Barman} and \textit{A. Dow} introduced the notion of strategic selective separability (abbreviated as \(SS^+\)) in [Topol.\ Proc.\ 37, 181--204 (2011; Zbl 1207.54036)] and that of strategic fan tightness in [Topology Appl. 159, No. 3, 806--813 (2012; Zbl 1246.54031)]. In this paper, the author proves that \(PR(X)\) is \(SS^+\) if and only if \(X\) is countable and every finite power of \(X\) has strategic fan tightness for finite sets. To show this, he proves that \(PR(X)\) has strategic fan tightness if and only if every finite power of \(X\) has strategic fan tightness for finite sets. Moreover, he gives examples of selectively separable Pixley-Roy hyperspaces which are not \(SS^+\) and of \(SS^+\) Pixley-Roy hyperspaces of uncountable \(\pi\)-weight.
scientific article; zbMATH DE number 6121487