Families of continuous retractions and function spaces

TL;DR

This paper investigates families of continuous retractions with rich properties, demonstrating how certain skeletons are preserved under space operations and introducing new concepts like q-skeletons to analyze function spaces.

Contribution

It introduces and studies new types of skeletons, solves an open question about Alexandroff duplicates, and characterizes spaces with strong r-skeletons using topological methods.

Findings

Alexandroff duplicate of a space with a full r-skeleton also has a full r-skeleton

Every compact subspace of C_p(X) is Corson if X has a full q-skeleton

Spaces with a strong r-skeleton are characterized as monotonically Sokolov

Abstract

In this article, we mainly study certain families of continuous retractions ($r$-skeletons) having certain rich properties. By using monotonically retractable spaces we solve a question posed by R. Z. Buzyakova in \cite{buz} concerning the Alexandroff duplicate of a space. Certainly, it is shown that if the space $X$ has a full $r$-skeleton, then its Alexandroff duplicate also has a full $r$-skeleton and, in a very similar way, it is proved that the Alexandroff duplicate of a monotonically retractable space is monotonically retractable. The notion of $q$-skeleton is introduced and it is shown that every compact subspace of $C_p(X)$ is Corson when $X$ has a full $q$-skeleton. The notion of strong $r$-skeleton is also introduced to answer a question suggested by F. Casarrubias-Segura and R. Rojas-Hern\'andez in their paper \cite{cas-rjs} by establishing that a space $X$ is monotonically Sokolov iff it is monotonically $\omega$-monolithic and has a strong $r$-skeleton. The techniques used here allow us to give a topological proof of a result of I. Bandlow \cite{ban} who used elementary submodels and uniform spaces.