Locally connected space

From Encyclopedia of Mathematics
Revision as of 17:13, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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).



[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:
This article was adapted from an original article by S.A. Bogatyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article