# Locally connected space

A topological space $X$ such that for any point $x$ and any neighbourhood $O _ {x}$ of it there is a smaller connected neighbourhood $U _ {x}$ of $x$. Any open subset of a locally connected space is locally connected. Any connected component of a locally connected space is open-and-closed. A space $X$ is locally connected if and only if for any family $\{ A _ {t} \}$ of subsets of $X$,

$$\partial \cup _ { t } A _ {t} \subset \ {\cup _ { t } {\partial A _ {t} } } bar$$

(here $\partial B$ is the boundary of $B$ and $\overline{B}\;$ is the closure of $B$). Any locally path-connected space is locally connected. A partial converse of this assertion is the following: Any complete metric locally connected space is locally path-connected (the Mazurkiewicz–Moore–Menger theorem).