$r$-skeletons on the Alexandroff duplicate

TL;DR

This paper explores the properties of $r$-skeletons on compact spaces, characterizing when they extend to the Alexandroff duplicate and analyzing their behavior on zero-dimensional spaces.

Contribution

It provides a characterization of compact spaces with extendable $r$-skeletons to their Alexandroff duplicates and examines the closure properties of $r$-skeletons on zero-dimensional spaces.

Findings

Characterization of spaces with extendable $r$-skeletons to Alexandroff duplicates.

Existence of an $r$-skeleton element with non-countable closure in zero-dimensional spaces.

Extension conditions for $r$-skeletons on compact spaces.

Abstract

An $r$-skeleton on a compact space is a family of continuous retractions having certain rich properties. The $r$-skeletons have been used to characterized the Valdivia compact spaces and the Corson compact spaces. Here, we characterized a compact space with an $r$-skeleton, for which the given $r$-skeleton can be extended to an $r$-skeleton on the Alexandroff Duplicate of the given space. Besides, we prove that if $X$ is a zero-dimensional compact space without isolated points and $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$, then there is $s\in \Gamma$ such that $cl(r_s[X])$ is not countable.