Difference between revisions of "Locally connected space"
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| − | + | 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|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==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.L. Kelley, "General topology" , v. Nostrand (1955) pp. 61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Čech, "Topological spaces" , Interscience (1966) pp. §21B</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.L. Kelley, "General topology" , v. Nostrand (1955) pp. 61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Čech, "Topological spaces" , Interscience (1966) pp. §21B</TD></TR></table> | ||
Latest revision as of 22:17, 5 June 2020
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=15600
Locally connected space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_connected_space&oldid=15600
This article was adapted from an original article by S.A. Bogatyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article