Difference between revisions of "Indecomposable continuum"
From Encyclopedia of Mathematics
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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 | + | 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 | + | 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> | + | <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> | ||
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=11594
Indecomposable continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indecomposable_continuum&oldid=11594
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