# Locally connected

A topological space $(X,\tau)$ is called locally connected at $x \in X$ if for every $V \in \tau$ with $x \in V$ there corresponds a connected open set $U \in \tau$ with $x \in U \subset V$. We say that $(X,\tau)$ is locally connected if for all $x \in X$, $(X,\tau)$ is locally connected at $x$.