Topological spaces with a $\sigma$-point finite base

Abstract:

The principal results of the paper are as follows. A topological space with a $\sigma$-point finite base has a $\sigma$-disjoint base if it is either hereditarily collectionwise normal or hereditarily screenable. From a metrization theorem of Arhangel’skiĭ, it follows that a ${T_1}$-space with a $\sigma$-point finite base is metrizable iff it is perfectly normal and collectionwise normal. A topological space with a $\sigma$-point base is quasi-developable in the sense of Bennett. Consequently a theorem of Čoban follows that for a topological space $(X,\Im )$ the following are equivalent: (a) $(X,\Im )$ is a metacompact normal Moore space, (b) $(X,\Im )$ is a perfectly normal ${T_1}$-space with a $\sigma$-point finite base.