Polish Space - Characterization

Characterization

There are numerous characterizations that tell when a second countable topological space is metrizable, such as Urysohn's metrization theorem. The problem of determining whether a metrizable space is completely metrizable is more difficult. Topological spaces such as the open unit interval (0,1) can be given both complete metrics and incomplete metrics generating their topology.

There is a characterization of complete separable metric spaces in terms of a game known as the strong Choquet game. A separable metric space is completely metrizable if and only if the second player has a winning strategy in this game.

A second characterization follows from Alexandrov's theorem. It states that a separable metric space is completely metrizable if and only if it is a subset of its completion in the original metric.