A note on joint metrizability of spaces on families of subspaces

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 FS(X)FS(X)={S∪{xS}{S∪{xS}: S is a convergent sequence of X   which converges to the point xS}xS}. A subspace Y of X is called almost discrete if Y   has at most one non-isolated point. Denote FAD(X)={Y:Y is an almost discrete subspace of X}FAD(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 FS(X)FS(X) (FAD(X)FAD(X)). We get some conclusions on JSM-spaces and JADM-spaces.A space X is said to be weak Fréchet if for any σ-closed discrete subspace B of X   with x∈B¯ there is a sequence {xn:n∈N}⊂B{xn:n∈N}⊂B which converges to x. We show that if X   is a Hausdorff weak Fréchet compactly metrizable strong Σ⁎Σ⁎-space then X has a σ-locally finite network.