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Difference between revisions of "Irreducible continuum"

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A non-degenerate [[Continuum|continuum]] that is irreducible between a certain pair of points, that is, does not contain any proper subcontinuum containing these points.
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A non-degenerate [[Continuum|continuum]] that is irreducible between a certain pair of points, that is, does not contain any proper subcontinuum containing these points.
  
 
====Comments====
 
====Comments====
An example is the famous curve of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052580/i0525801.png" />: it is the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052580/i0525802.png" /> of the plane. This curve is irreducible between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052580/i0525803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052580/i0525804.png" />.
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An example is the famous curve of $  \sin ( 1 / x ) $:  
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it is the subset $  \{ 0 \} \times [ - 1 , 1 ] \cup \{ {( x, \sin ( 1 / x ) ) } : {0 < x \leq  1 } \} $
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of the plane. This curve is irreducible between the points $  ( 0 , 0 ) $
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and $  ( 1 , \sin  1 ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''2''' , Acad. Press  (1968)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''2''' , Acad. Press  (1968)  (Translated from French)</TD></TR></table>

Latest revision as of 22:13, 5 June 2020


A non-degenerate continuum that is irreducible between a certain pair of points, that is, does not contain any proper subcontinuum containing these points.

Comments

An example is the famous curve of $ \sin ( 1 / x ) $: it is the subset $ \{ 0 \} \times [ - 1 , 1 ] \cup \{ {( x, \sin ( 1 / x ) ) } : {0 < x \leq 1 } \} $ of the plane. This curve is irreducible between the points $ ( 0 , 0 ) $ and $ ( 1 , \sin 1 ) $.

References

[a1] K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French)
How to Cite This Entry:
Irreducible continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_continuum&oldid=15992
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article