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Difference between revisions of "Locally connected space"

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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603401.png" /> such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603402.png" /> and any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603403.png" /> of it there is a smaller connected neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603404.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603405.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603406.png" /> is locally connected if and only if for any family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603407.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603408.png" />,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603409.png" /></td> </tr></table>
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(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l06034010.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l06034011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l06034012.png" /> is the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l06034013.png" />). 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).
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A topological space  $  X $
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such that for any point  $  x $
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and any neighbourhood  $  O _ {x} $
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of it there is a smaller connected neighbourhood  $  U _ {x} $
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of $  x $.  
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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 $
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is locally connected if and only if for any family  $  \{ A _ {t} \} $
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of subsets of  $  X $,
  
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$$
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\partial  \cup _ { t } A _ {t}  \subset  \
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{\cup _ { t } {\partial  A _ {t} } } bar
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$$
  
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(here  $  \partial  B $
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is the boundary of  $  B $
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and  $  \overline{B}\; $
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is the closure of  $  B $).
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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=47691
This article was adapted from an original article by S.A. Bogatyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article