# Dispersive space

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 54D05 [MSN][ZBL]

*hereditarily disconnected space*

A topological space which contains no connected sets with more than one point.

#### Comments

Some authors call spaces as above totally disconnected. However, a space is commonly called totally disconnected if for all $x\neq y$ in $X$ there is a closed and open set $C$ such that $x\in C$ and $y\notin C$. A closed and open set is also called a clopen set.

See also Totally-disconnected space.

#### References

[a1] | K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) (Translated from French) |

[a2] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |

**How to Cite This Entry:**

Dispersive space.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Dispersive_space&oldid=35756

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