I have seen two a priori different definitions of locally connected space :
1) For all point $x$, and neighborhood $V$ of $x$, there is a connected neighborhood $C$ of $x$ such that $C\subseteq V$
2) For all point $x$, and open set $U$ containing $x$, there is an open connected neighborhood $O$ of $x$ such that $O\subseteq U$.
I imagine those two definitions are identical, but I don't see why.
The same thing for locally path-connected spaces, we can either use open neighborhoods or just neighborhoods, are the two definitions the same?