# Locally connected space

From Encyclopedia of Mathematics

A topological space such that for any point and any neighbourhood of it there is a smaller connected neighbourhood of . 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 is locally connected if and only if for any family of subsets of ,

(here is the boundary of and is the closure of ). 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. S.A. Bogatyi (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Locally_connected_space&oldid=15600

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098