A note on Collins-Roscoe Structuring Mechanism

TL;DR

This paper investigates the countable $(F)$-property in function spaces, providing conditions under which it holds, especially for Corson compact spaces, and explores its preservation under certain product operations.

Contribution

It establishes new sufficient conditions for $C_p(X)$ to have the countable $(F)$-property and proves its preservation under $ ext{Sigma}_s$-products, answering open questions.

Findings

$C_p(X)$ has countable $(F)$-property for Corson compact $X$

Countable $(F)$-property is preserved under $ ext{Sigma}_s$-products

Identifies classes of function spaces with hereditarily $D$-property

Abstract

A space $X$ has countable $(F)$-property if it has countable point network satisfying the Collins-Roscoe structuring mechanism. Some sufficient conditions for $C_p(X)$ having countable $(F)$-property are obtained. As a corollary, we prove that if $X$ is Corson compact, $C_p(X)$ satisfies countable $(F)$. This answers a question raised by Tkachuk. Also we get a class of function spaces with hereditarily $D$-property. We also prove that the countable $(F)$-property is preserved by taking $\Sigma_s$-product. This answers questions of Tkachuk positively.