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Locally connected space

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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