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).

References

[a1]	G.L. Kelley, "General topology" , v. Nostrand (1955) pp. 61
[a2]	E. Čech, "Topological spaces" , Interscience (1966) pp. §21B