Dispersive space

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

hereditarily disconnected space

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


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.


[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)
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Dispersive space. Encyclopedia of Mathematics. URL:
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