In C.C.Pugh's book "Real Mathematical Analysis" a metric space $X$ is said to be locally path-connected if for each $p\in X$ and (open) neighborhood $U$ containing $p$ there is a path-connected (open) subneighborhood $V$ of $p$.
However, Wikipedia states that a topological space $X$ (for our purposes, let's assume it is a metric space) is locally path-connected if for every open set $U$ containing $x$, there exists a path connected open set $V$ with $x\in V\subseteq U$.
The former definition gives the sense that the neighborhood $U$ and the subneighborhood $V$ are open balls around $p$ (i.e. they are equally distributed around $p$), but the latter definition doesn't. How do I show that both the definitions are equivalent?