Knaster continuum

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.