Seeking a network characterization of Corson compacta

Abstract

We say that a collection $\mathcal{A}$ of subsets of $X$ has property $(CC)$ if there is a set $D$ and point-countable collections $\mathcal{C}$ of closed subsets of $X$ such that for any $A\in \mathcal{A}$ there is a finite subcollection $\mathcal{F}$ of $\mathcal{C}$ such that $A=D\setminus \bigcup \mathcal{F}$. Then we prove that any compact space is Corson if and only if it has a point-$\sigma$-$(CC)$ base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in ``A Biased View of Topology as a Tool in Functional Analysis'' (2014) by B. Cascales and J. Orihuela and as in ``Network characterization of Gul'ko compact spaces and their relatives'' (2004) by F. Garcia, L. Oncina, J. Orihuela, which asked whether there is a network characterization of the class of Corson compacta.