Metrization theorems

A metrizable space istopological space thathomeomorphic tometric space. Metrization theoremstheorems that give sufficient conditions fortopological spacebe metrizable.

For explanationsmany ofterms usedthis article,reader should seetopology glossary.

Metrizable spaces inherit all topological properties from metric spaces. For example, theyHausdorff paracompact spaces (and hence normalTychonoff)first countable.

The first really useful metrization theorem was Urysohn's metrization theorem. This states that every second-countable regular Hausdorff spacemetrizable. So,example, every second-countable manifoldmetrizable. (Historical note: The form oftheorem shown here wasfact proved by Tychonoff1926. What Urysohn had shown, inpaper published posthumously1925, wasslightly weaker result that every second-countable normal Hausdorff spacemetrizable.)

Several other metrization theorems follow as simple corollariesUrysohn's Theorem. For example,compact Hausdorff spacemetrizable ifonly if itsecond-countable.

Urysohn's Theorem can be restated as: A topological spaceseparablemetrizable ifonly if itsecond-countable, regularHausdorff. The Nagata-Smirnov metrization theorem extends this tonon-separable case. It states thattopological spacemetrizable ifonly if itregularHausdorffhasσ-locally finite base. A σ-locally finite base isbase which isunioncountably many locally finite collectionsopen sets.

A spacesaidbe locally metrizable if every point hasmetrizable neighbourhood. Smirnov proved thatlocally metrizable Hausdorff spacemetrizable ifonly if itparacompact. In particular,manifoldmetrizable ifonly if itparacompact.