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''hereditarily indecomposable continuum''
 
''hereditarily indecomposable continuum''
  
A continuum each subcontinuum of which is indecomposable. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055530/k0555301.png" /> is called indecomposable if it is connected and cannot be represented as the union of two closed connected proper subsets of it.
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A [[continuum]] each subcontinuum of which is [[Indecomposable continuum|indecomposable]]. A space is called indecomposable if it is connected and cannot be represented as the union of two closed connected proper subsets of it.
  
The first proof of the existence of such a continuum was given by B. Knaster [[#References|[1]]]. In the space of all subcontinua of the ordinary square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055530/k0555302.png" />, the set of all Knaster continua is an everywhere-dense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055530/k0555303.png" />-set [[#References|[2]]].
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The first proof of the existence of such a continuum was given by B. Knaster [[#References|[1]]]. In the space of all subcontinua of the ordinary square I^2, the set of all Knaster continua is an [[Everywhere-dense set|everywhere-dense]] [[G-delta|G_\delta]]-set [[#References|[2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Knaster,  "Un continu dont tout sous-continu est indécomposable"  ''Fund. Math.'' , '''3'''  (1922)  pp. 247–286</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Mazurkiewicz,  "Sur les continus absolument indécomposables"  ''Fund. Math.'' , '''16'''  (1930)  pp. 151–159</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  B. Knaster,  "Un continu dont tout sous-continu est indécomposable"  ''Fund. Math.'' , '''3'''  (1922)  pp. 247–286 {{ZBL|48.0212.01}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  S. Mazurkiewicz,  "Sur les continus absolument indécomposables"  ''Fund. Math.'' , '''16'''  (1930)  pp. 151–159 {{ZBL|56.1135.02}}</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
See also [[Pseudo-arc|Pseudo-arc]] and [[Hereditarily indecomposable continuum|Hereditarily indecomposable continuum]].
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See also [[Pseudo-arc]] and [[Hereditarily indecomposable continuum]].

Latest revision as of 16:42, 10 January 2015

2020 Mathematics Subject Classification: Primary: 54F15 [MSN][ZBL]

hereditarily indecomposable continuum

A continuum each subcontinuum of which is indecomposable. A space X is called indecomposable if it is connected and cannot be represented as the union of two closed connected proper subsets of it.

The first proof of the existence of such a continuum was given by B. Knaster [1]. In the space of all subcontinua of the ordinary square I^2, the set of all Knaster continua is an everywhere-dense G_\delta-set [2].

References

[1] B. Knaster, "Un continu dont tout sous-continu est indécomposable" Fund. Math. , 3 (1922) pp. 247–286 Zbl 48.0212.01
[2] S. Mazurkiewicz, "Sur les continus absolument indécomposables" Fund. Math. , 16 (1930) pp. 151–159 Zbl 56.1135.02


Comments

See also Pseudo-arc and Hereditarily indecomposable continuum.

How to Cite This Entry:
Knaster continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Knaster_continuum&oldid=36191
This article was adapted from an original article by L.G. Zambakhidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article