Families of retractions and families of closed subsets on compact spaces

TL;DR

This paper explores the structure of $r$-skeletons and introduces $ ho$-skeletons, establishing their relationship with Valdivia and Corson compact spaces, and providing new characterizations and extension conditions.

Contribution

It introduces the concept of $ ho$-skeletons, generalizes $ ext{Sigma}$-products, and proves the equivalence between retractional-skeletons and $ ho$-skeletons in compact spaces.

Findings

A zero-dimensional compact space with countable images has a dense set of isolated points.

Conditions are given for extending $r$-skeletons to Alexandroff Duplicates.

A compact space admits a retractional-skeleton iff it admits a $ ho$-skeleton.

Abstract

It is know that the Valdivia compact spaces can be characterized by a special family of retractions called $r$-skeleton (see \cite{kubis1}). Also we know that there are compact spaces with $r$-skeletons which are not Valdivia. In this paper, we shall study $r$-squeletons and special families of closed subsets of compact spaces. We prove that if $X$ is a zero-dimensional compact space and $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$ such that $|r_s(X)| \leq \omega$ for all $s\in \Gamma$, then $X$ has a dense subset consisting of isolated points. Also we give conditions to an $r$-skeleton in order that this $r$-skeleton can be extended to an $r$-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a $\Sigma$-product of a power of the unit interval. Following this fact we introduce the notion of $\pi$-skeleton on a compact space $X$ by embedding $X$ in a suitable power of the unit interval together with a pair $(\mathcal{F},\varphi)$, where $\mathcal{F}$ is family of metric separable subspaces of $X$ and $\varphi$ an $\omega$-monotone function which satisfy certain properties. This new notion generalize the idea of a $\Sigma$-product. We prove that a compact space admits a retractional-skeleton iff it admits a $\pi$-skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with retractional-skeletons admits an retractional-skeleton (see \cite{cuth1}). In \cite{casa1}, the Corson compact spaces are characterized by a special family of closed subsets. Following this direction, we introduce the notion of weak $c$-skeleton which under certain conditions characterizes the Valdivia compact spaces and compact spaces with $r$-skeletons.