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Difference between revisions of "Locally connected continuum"

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A continuum that is a [[Locally connected space|locally connected space]]. Examples of locally connected continua are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060330/l0603301.png" />-dimensional cube, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060330/l0603302.png" /> the [[Hilbert cube|Hilbert cube]], and all Tikhonov cubes (cf. [[Tikhonov cube|Tikhonov cube]]). The union of the graph of the function
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and the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060330/l0603304.png" /> gives an example of a continuum that is not locally connected (at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060330/l0603305.png" />). A metrizable continuum is locally connected if and only if it is a curve in the sense of Jordan (cf. [[Line (curve)|Line (curve)]]). Any metrizable locally connected continuum is path-connected (cf. [[Path-connected space|Path-connected space]]). Moreover, any two distinct points of such a continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060330/l0603306.png" /> are contained in a simple arc lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060330/l0603307.png" />.
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A continuum that is a [[Locally connected space|locally connected space]]. Examples of locally connected continua are the  $  n $-
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dimensional cube,  $  n = 0 , 1 \dots $
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the [[Hilbert cube|Hilbert cube]], and all Tikhonov cubes (cf. [[Tikhonov cube|Tikhonov cube]]). The union of the graph of the function
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$$
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y  =  \sin 
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\frac{1}{x}
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,\ \
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0 < x \leq  1 ,
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$$
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and the interval $  I = \{ {( 0 , y ) } : {- 1 \leq  y \leq  1 } \} $
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gives an example of a continuum that is not locally connected (at the points of $  I $).  
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A metrizable continuum is locally connected if and only if it is a curve in the sense of Jordan (cf. [[Line (curve)|Line (curve)]]). Any metrizable locally connected continuum is path-connected (cf. [[Path-connected space|Path-connected space]]). Moreover, any two distinct points of such a continuum $  K $
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are contained in a simple arc lying in $  K $.

Latest revision as of 22:17, 5 June 2020


A continuum that is a locally connected space. Examples of locally connected continua are the $ n $- dimensional cube, $ n = 0 , 1 \dots $ the Hilbert cube, and all Tikhonov cubes (cf. Tikhonov cube). The union of the graph of the function

$$ y = \sin \frac{1}{x} ,\ \ 0 < x \leq 1 , $$

and the interval $ I = \{ {( 0 , y ) } : {- 1 \leq y \leq 1 } \} $ gives an example of a continuum that is not locally connected (at the points of $ I $). A metrizable continuum is locally connected if and only if it is a curve in the sense of Jordan (cf. Line (curve)). Any metrizable locally connected continuum is path-connected (cf. Path-connected space). Moreover, any two distinct points of such a continuum $ K $ are contained in a simple arc lying in $ K $.

How to Cite This Entry:
Locally connected continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_connected_continuum&oldid=47690
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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