A characterization of the compact spaces with a retractional skeleton

Abstract

In [14] the authors introduced the concept of retractional skeleton (r-skeleton) and they characterized the Valdivia compact spaces with this notion. In [2], the authors introduced the notion of (full) c-skeleton which is a pair consisting of a family of closed subsets and an ω-monotone map which satisfy certain properties; and they proved that a compact space is a Corson compact space iff it admits a full c-skeleton. The main purpose of this article is to prove that a compact space admits a retractional skeleton iff it admits a c-skeleton. Moreover, we establish a condition on a c-skeleton which allows a compact space be a Valdivia compact space. Additionally, we provide an answer to Question 5.8 of the paper [2] by using an argument different from the one used in [11] to respond to the same question.