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Difference between revisions of "Indecomposable continuum"

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Two equivalent definitions: 1) there are three points such that the continuum is irreducible between each pair of points from these three (cf. [[Irreducible continuum|Irreducible continuum]]); and 2) every proper subcontinuum is nowhere-dense.
 
Two equivalent definitions: 1) there are three points such that the continuum is irreducible between each pair of points from these three (cf. [[Irreducible continuum|Irreducible continuum]]); and 2) every proper subcontinuum is nowhere-dense.
  
In indecomposable continua one has composants, which are like components: the composant of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050520/i0505201.png" /> is the union of all proper subcontinua containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050520/i0505202.png" />.
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In indecomposable continua one has composants, which are like components: the composant of a point $x$ is the union of all proper subcontinua containing $x$.
  
Examples of indecomposable continua are the [[Pseudo-arc|pseudo-arc]], which is even a [[Hereditarily indecomposable continuum|hereditarily indecomposable continuum]]; a [[Solenoid|solenoid]]; and the remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050520/i0505203.png" /> in the [[Stone–Čech compactification|Stone–Čech compactification]] of the half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050520/i0505204.png" />.
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Examples of indecomposable continua are the [[Pseudo-arc|pseudo-arc]], which is even a [[Hereditarily indecomposable continuum|hereditarily indecomposable continuum]]; a [[Solenoid|solenoid]]; and the remainder $\beta\mathcal{H} \setminus \mathcal{H}$ in the [[Stone–Čech compactification|Stone–Čech compactification]] of the half-line $\mathcal{H} = [0,\infty)$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.P. Bellamy,  "A non-metric indecomposable continuum"  ''Duke Math. J.'' , '''38'''  (1971)  pp. 15–20</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''2''' , Acad. Press  (1968)  (Translated from French)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  D.P. Bellamy,  "A non-metric indecomposable continuum"  ''Duke Math. J.'' , '''38'''  (1971)  pp. 15–20</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''2''' , Acad. Press  (1968)  (Translated from French)</TD></TR>
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</table>

Revision as of 18:19, 1 October 2013

A non-degenerate continuum that cannot be represented as the union of two proper subcontinua.


Comments

Two equivalent definitions: 1) there are three points such that the continuum is irreducible between each pair of points from these three (cf. Irreducible continuum); and 2) every proper subcontinuum is nowhere-dense.

In indecomposable continua one has composants, which are like components: the composant of a point $x$ is the union of all proper subcontinua containing $x$.

Examples of indecomposable continua are the pseudo-arc, which is even a hereditarily indecomposable continuum; a solenoid; and the remainder $\beta\mathcal{H} \setminus \mathcal{H}$ in the Stone–Čech compactification of the half-line $\mathcal{H} = [0,\infty)$.

References

[a1] D.P. Bellamy, "A non-metric indecomposable continuum" Duke Math. J. , 38 (1971) pp. 15–20
[a2] K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French)
How to Cite This Entry:
Indecomposable continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indecomposable_continuum&oldid=30592
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