locally connected

A topological space $X$ is locally connected at a point $x \in X$ if every neighborhood $U$ of $x$ contains a connected neighborhood $V$ of $x$ . The space $X$ is locally connected if it is locally connected at every point $x \in X$ .

A topological space $X$ is locally path connected at a point $x \in X$ if every neighborhood $U$ of $x$ contains a path connected neighborhood $V$ of $x$ . The space $X$ is locally path connected if it is locally path connected at every point $x \in X$ .