Locally connected space
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|
Locally connected space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_connected_space&oldid=15600