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We say that a collection $𝒜$ of subsets of $X$ has property $\left(CC\right)$ if there is a set $D$ and point-countable collections $𝒞$ of closed subsets of $X$ such that for any $A\in 𝒜$ there is a finite subcollection $ℱ$ of $𝒞$ such that $A=D\setminus \bigcup ℱ$. Then we prove that any compact space is Corson if and only if it has a point-$\sigma$-$\left(CC\right)$ 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...

Necessary conditions and sufficient conditions are given for $C_p\left(X\right)$ to be a (σ-) m₁- or m₃-space. (A space is an m₁-space if each of its points has a closure-preserving local base.) A compact uncountable space K is given with $C_p\left(K\right)$ an m₁-space, which answers questions raised by Dow, Ramírez Martínez and Tkachuk (2010) and Tkachuk (2011).