# Isolated point

From Encyclopedia of Mathematics

(Redirected from Dense-in-itself set)

*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*.

#### References

[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 |

**How to Cite This Entry:**

Dense-in-itself set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dense-in-itself_set&oldid=42517