# Knaster continuum

From Encyclopedia of Mathematics

2010 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.

**How to Cite This Entry:**

Knaster continuum.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Knaster_continuum&oldid=36191

This article was adapted from an original article by L.G. Zambakhidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article