Selectivity properties of spaces

TL;DR

This paper explores the selectivity properties of certain topological spaces, providing new examples and counterexamples under various set-theoretic assumptions, advancing understanding of continuous selections.

Contribution

It constructs novel examples of spaces with specific selectivity properties, including spaces that are L-selective but not Q-selective, under different set-theoretic assumptions.

Findings

Constructed an L-selective space not Q-selective from =

Produced an L-selective space not selective for a P-point ultrafilter under CH

Provided ZFC examples of Fre9chet spaces with countable subsets that are not L-selective

Abstract

This paper addresses several questions of Feng, Gruenhage, and Shen which arose from Michael's theory of continuous selections from countable spaces. We construct an example of a space which is $L$-selective but not $\mathbb{Q}$-selective from $\mathfrak{d}=\omega_1$, and an $L$-selective space which is not selective for a $P$-point ultrafilter from the assumption of $\mathsf{CH}$. We also produce $\mathsf{ZFC}$ examples of Fr\'echet spaces where countable subsets are first countable which are not $L$-selective.