Indecomposable continuum

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)