Isolated point

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of a subspace $A$ of a topological space $X$

A point $a\in A$ such that the intersection of some neighbourhood of $a$ with $A$ consists of the point $a$ alone.

A subset $A$ with no isolated points is dense-in-itself; a closed dense-in-itself subset is a perfect set.


[1] Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (second edition). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3 MR507446 Zbl 0386.54001
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Isolated point. 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