G-delta criterion for topological completeness
= G-delta criterion for topological completeness
{c}
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.