Colloquium: Vladimir Tkachuk

Speaker: Dr. Vladimir Tkachuk, Universidad Autónoma Metropolitana de México

Title: Properties of $C_p(X)$ determined by its discrete subspaces 

Abstract:

Given a topological property $P$, say that a space $Z$ is discretely $P$ if the closure of $D$ in $Z$ has $P$ whenever $D$ is a discrete subspace of $Z$. We study the properties $P$ in $C_p(X)$ that are equivalent to $C_p(X)$ being discretely $P$. We will show that it is independent of ZFC whether discrete metrizability of $C_p(X)$ implies its metrizability for a compact space X. We also establish that it is consistent with ZFC that countable tightness and Lindelof $\Sigma$-property are not discretely reflexive in spaces $C_p(X)$. However, if $C_p(X)$ is discretely Cech-complete, then $X$ is countable and discrete. If $C_p(X)$ is discretely $\sigma$-compact, then $X$ has to be finite.