Supercomplete topological spaces (Q2460655)

Completeness in some form or another is always desirable: it guarantees that certain processes will converge. An important completeness property is Čech-completeness: the space is a \(G_\delta\)-set in its Čech-Stone compactification. The authors define a stronger property, called supercompleteness: there is a sequence \(\langle\mathcal{U}_n\rangle_n\) of open covers such that every filter~\(\mathcal{F}\), with the property that for every~\(n\) the intersection \(\bigcap\{\operatorname{St}(F,\mathcal{U}_n:F\in\mathcal{F}\}\) belongs to~\(\mathcal{F}\), has a cluster point. Supercompleteness is equivalent to Čech-completeness in the class of paracompact spaces and also in the class of topological groups. Supercompleteness is an inverse invariant of perfect maps and an invariant of open perfect maps. The ordinal space~\(\omega_1\) is locally compact but not supercomplete. The authors introduce variants of supercompleteness by considering only filters of closed sets or filters with a countable base. All are equivalent to Čech-completeness in the class of metrizable spaces.