I have read two definitions of a topological space $X$ being locally connected at a point $x \in X$ , and want to prove they are equivalent.
Def.1 (wiki) A topological space $X$ is locally connected at $x \in X$ if for every open set $U$ with $x \in U$ there exist an open connected set $V \subseteq U$ with $x \in V$
Def.2 (wikibooks) A topological space $X$ is locally connected at $x \in X$ if for every neighbourhood $U$ of $x$ there exist a connected neighbourhood $V$ of $x$ with $V \subseteq U$.
I like more def.2 as it seems simplier without having to mention open sets, it just mention neighbourhoods. Now I'll try to prove they are equivalent:
def.1 $\Rightarrow$ def.2) As every neighbourhood $U$ of $x$ contains an open set $U'$ containing $x$, and by def.1 there is an open connected set $V' \subseteq U'$ with $x \in V'$ and $V = V'$ is a connected neighbourhood of $x$ with $V \subseteq U$.
def.2 $\Rightarrow$ def.1) (Here comes the problem) Let $U$ be an open set with $x \in U$, so it is a neighbourhood of $x$, and by def.2 there exists $V \subseteq U$ connected with $x \in V$. The problem is that $V$ could not be open. As it is a neighbourhood of $x$, it contains at least one open subset $V' \subseteq$ with $x \in V'$. But now $V'$ could not be connected. I trying choosing the connected component $C_x$ of $V'$ that contains $x$. I know that $C_x$ is closed, and connected, but I think it could be not open. And I don't know what else to do.
Are these definitions really equivalent? If yes I would appreciate any help finishing the proof, and if no I would appreciate a counterexample: A space $X$ with a point $x \in X$ that is def.2 but is not def.1