Basis properties of topologies compatible with (not necessarily symmetric) distance-functions

Abstract

One of the aims of this paper is to generalize a theorem of P. Fletcher and W.F. Lindgren characterizing second countable spaces. On behalf of that, we investigate a basis property related to the concept of σ-Q-bases defined by Fletcher and Lindgren and orthobases studied by P. Nyikos and W.F. Lindgren. In this new setting we state a necessary and sufficient condition for ω_μ-quasimetrizability of topological spaces and we discuss a problem of P. Fletcher and W.F. Lindgren and a related theorem of S. Nedev concerning quasimetrizability of T₁-spaces. As a corollary we give a characterization of ω_μ-additive spaces having a base of cardinality ω_μ— In the second part of the paper, we study (not necessarily symmetric) distance-functions on a space X taking their values in a partially ordered group H. We show that every T₁-space X is quasimetrizable in this generalized sense.