Joint metrizability of subspaces and perfect mappings

Following a general idea in [6], [7], we introduce and study in this paper the concept of a JPM-space. We call in this way topological spaces admitting a metric which metrizes every metrizable subspace of X. It is shown that any Hausdorff sequential JPM-space is metrizable. It is also proved that perfect mappings with metrizable fibers preserve the class of Hausdorff JPM-spaces. Some new results concerning countably metrizable spaces and compactly metrizable spaces introduced in [6] and [7] are also obtained.