Exercise 33, § 4

Let $X$ be a metric space such that every point x possesses a neighbourhood $U(x)$ which is a complete subspace for the metric of X. Now prove that there exists on X a metric compatible with the topology of X and for which X is a complete metric space. (If $\mathfrak{R}$ is

¹ One can prove that the conclusion still holds if one assumes only that R is closed; see E. Michael, Proc. Amer. Math. Soc., t. VIII (1957), p. 822–828.

the covering of X formed by the U(x) for x ∈ X, consider the neighbourhood V_R of the diagonal Δ in X × X, and use IX, p. 113, exerc. 26 a); by using IX, p. 6, prop. 2, define on X a metrizable uniform structure finer than the structure defined by the metric of X, compatible with the topology of X, and for which X is complete.)