I have found two different definitions of locally connectedness:
(1) A topological space $(X,\mathcal{T})$ is called ``locally connect at $x\in X$'', if every open neighbourhood $U$ of $x$ contains an open and connected neighbourhood $V$ of $x$.
(2) A topological space $(X,\mathcal{T})$ is called ``locally connect at $x\in X$'', if every neighbourhood $U$ of $x$ contains a connected neighbourhood $V$ of $x$.
So they differ only by the word ``open''. Are this two definition equivalent? If not, which one is the more standard one?
A neighbourhood of $x$ is defined as a set, which contains an open set, which contains $x$. So a neighbourhood is not necessarely open. That from definition (1) follows that (2) is fulfilled is obvious...but what is with the other way round?
The definitions are not equivalent. If there's a connected neighborhood, there's not necessarily a connected open set containing $x$ in the neighborhood. See "broom space". The second definition is known as weak local connectedness.