Definition 5.14.1 (Locally Path-Connected).label Let $X$ be a topological space, then $X$ is locally path-connected if for every $x \in X$, there exists a fundamental system of neighbourhoods consisting of path-connected sets at $x$.
Definition 5.14.1 (Locally Path-Connected).label Let $X$ be a topological space, then $X$ is locally path-connected if for every $x \in X$, there exists a fundamental system of neighbourhoods consisting of path-connected sets at $x$.