$r$-skeletons on the Alexandroff duplicate

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.