$G_\delta$ covers of compact spaces

TL;DR

This paper addresses a longstanding question in topology by constructing a compact space with a specific $G_\delta$ cover property and explores related properties in various classes of spaces, providing new bounds and results.

Contribution

It constructs a compact space with a $G_\delta$ cover lacking a continuum-sized dense subcollection and proves new $G_\delta$ cover properties in countably compact and Lindel"of spaces, refining known cardinality bounds.

Findings

Constructed a compact space with a $G_\delta$ cover with no continuum-sized dense subcollection.

Proved that in certain spaces, every $G_\delta$ cover has a continuum-sized subcover.

Refined bounds on the cardinality of homogeneous spaces with countable tightness.

Abstract

We solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_\delta$ cover with no continuum-sized ($G_\delta$)-dense subcollection. We also prove that in a countably compact weakly Lindel\"of normal space of countable tightness, every $G_\delta$ cover has a $\mathfrak{c}$-sized subcollection with a $G_\delta$-dense union and that in a Lindel\"of space with a base of multiplicity continuum, every $G_\delta$ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on the cardinality of homogeneous compacta of countable tightness.