Topological completeness

A metrizable space is topologically complete when some metric inducing its topology is complete. The given metric need not be complete: the open interval $(0,1)$ is an example. A Polish space additionally requires separability.

A subspace of a complete metric space admits a compatible complete metric if and only if it is a G-delta set. For the reverse direction, adjoining the coordinates $1/d(x,X\setminus U_n)$ to the ambient metric prevents Cauchy sequences from approaching any excluded closed boundary. For the forward direction, small-diameter relative open covers in a compatible complete metric produce the ambient open sets.