Absoluteness of $F_{σδ}$ property

Vojtěch Kovařík

Published 2016-07-13Version 1

In general topology, the class of \v{C}ech-complete spaces is well studied. A space $X$ is \v{C}ech-complete, if it is a $G_\delta$ subset of its \v{C}ech-Stone compactification $\beta X$, and this is equivalent to $X$ being a $G_\delta$ subset of its every compactification. In functional analysis, one naturally encounters many spaces which, when equipped with weak topology, are $F_{\sigma\delta}$ rather than $G_\delta$ subsets of some of their compactifications. For example, all separable Banach spaces (or more generally, all WCG Banach spaces) enjoy this property. However, this property is not "absolute" - a Banach space can be $F_{\sigma\delta}$ in some, but not all of its compactifications.\vknote{Possibly add a reference here} We study topological spaces which are $F_{\sigma\delta}$ in some of their compactifications, giving for each such $X$ a wider class of compactifications which all contain $X$ as an $F_{\sigma\delta}$ subset.