Polish Spaces

Abstract

Let (X, d) be a metric space. A Cauchy sequence is a sequence (x _n ) of elements of X such that lim _m,n d(x _m ,x _n ) = 0. We call (X,d) complete if every Cauchy sequence has a limit in X. Given any metric space (X,d), there is a complete metric space $\hat X,\hat d$ such that (X,d) is a subspace of $\hat X,\hat d$ and X is dense in $\hat X$. This space is unique up to isometry and is called the completion of (X,d). Clearly, $\hat X$ is separable iff X is separable.