A note on joint metrizability of spaces on families of subspaces

摘要

In this note, we introduce concepts of JSM-spaces and JADM-spaces following a general idea of Arhangel'skii and Shumrani. Let X be a topological space. Denote F-s(X)={S boolean OR {x(s)}: S is a convergent sequence of X which converges to the point x(S)}. A subspace Y of X is called almost discrete if Y has at most one non-isolated point. Denote F-AD(X) = {Y : Y is an almost discrete subspace of X}. A space X is called a JSM-space (JADM-space) if there is a metric d on the set X such that d metrizes all subspaces of X which belong to F-s(X) (F-AD(X)). We get some conclusions on JSM-spaces and JADM-spaces.