Corollary 2

A locally compact, locally connected and metrizable topological space is locally arcwise connected. If it is connected, it is arcwise connected.

It suffices to prove the first assertion (III, p. 260, prop. 7). Let X be a locally compact and locally connected metric space. The open, connected and relatively compact subsets of X constitute a basis of the topology of X. Let U be such a set; since U is an open subset of its closure, which is a compact metric space, hence complete, there exists according to Lemma 2 a distance on U compatible with its topology for which U is complete. It follows from prop. 11 of III, p. 264 that U is arcwise connected, whence the corollary.