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Indecomposable continuum

From Encyclopedia of Mathematics
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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=31022
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