Characterizing Corson and Valdivia compact spaces

Abstract

Abstract. We give a new characterization of Valdivia compact spaces: A compact space is Valdivia if and only if it has a dense commutatively monotonically retractable subspace. This result solves Problem 5.12 from [6]. Besides, we introduce the notion of full c-skeleton and prove that a compact space is Corson if and only if it has a full c-skeleton.

Characterization of (semi-)Eberlein compacta using retractional skeletons

“…This part is essentially known as similar results were obtained e.g. in [5,Lemma 2.5] (using other methods than suitable models), but our approach is in a certain sense much more flexible (most importantly, because it may be combined with other statements involving suitable models) and we actually use this flexibility later. As a corollary of our investigations we show in Theorem 21 that we may in a certain way combine properties of countably many retractional skeletons.…”

“…Let us note that, quite surprisingly, there was independently introduced also the notion of monotonically retractable topological spaces which turned out to be very closely related to the study of compact spaces that admit a retractional skeleton, see [10], and from there on, several results and modifications of the corresponding notions were considered, see e.g. [5,16,17].…”

“…In Section 4, inspired by the proof of [5,Theorem 2.6], we aim at seeing as concretely as possible the "Valdivia embedding" of compact spaces with a commutative retractional skeleton. As a consequence we obtain the following result which might be thought of as the third main result of the whole paper.…”

Characterization of (semi-)Eberlein compacta using retractional skeletons

“…Quite surprisingly, also the notion of monotonically retractable topological spaces was independently introduced, which turned out to be very closely related to the study of compact spaces that admit a retractional skeleton (see [11]), and from there on, several results and modifications of the corresponding notions were considered (see e.g. [5,17,18]).…”

“…In Section 4, inspired by the proof of [5,Theorem 2.6], we aim at seeing as concretely as possible the "Valdivia embedding" of compact spaces with a commutative retractional skeleton. As a consequence we obtain the following result which might be thought of as the fourth main result of the whole paper.…”