# 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).

#### Comments

#### References

[a1] | G.L. Kelley, "General topology" , v. Nostrand (1955) pp. 61 |

[a2] | E. Čech, "Topological spaces" , Interscience (1966) pp. §21B |

**How to Cite This Entry:**

Locally connected space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Locally_connected_space&oldid=47691