Namespaces
Variants
Actions

Difference between revisions of "Locally connected continuum"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
−
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
+
<!--
 +
l0603301.png
 +
$#A+1 = 7 n = 0
 +
$#C+1 = 7 : ~/encyclopedia/old_files/data/L060/L.0600330 Locally connected continuum
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
−
<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/l060330/l0603303.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
−
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" />.
+
A continuum that is a [[Locally connected space|locally connected space]]. Examples of locally connected continua are the  $  n $-
 +
dimensional cube,  $  n = 0 , 1 \dots $
 +
the [[Hilbert cube|Hilbert cube]], and all Tikhonov cubes (cf. [[Tikhonov cube|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)|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 $
 +
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=12579
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Locally_connected_continuum&oldid=47690"