Corollary 1

A locally connected topological space, whose topology can be defined by a distance for which it is complete, is locally arcwise connected.

Let X be such a space and let U be an open and connected subset of X. According to Lemma 2 below, there exists a distance on U compatible with its topology for which U is complete. Since U is locally connected, it follows from Proposition 11 that U is arcwise connected.