# Condensation point

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## Definition

In mathematics, a **condensation point** *p* of a subset *S* of a topological space, is any point *p*, such that every open neighborhood of *p* contains uncountably many points of *S*. Thus, according to the axiom of choice, "condensation point" is synonymous with "-accumulation point".{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## Examples

- If
*S*= (0,1) is the open unit interval, a subset of the real numbers, then 0 is a condensation point of*S*. - If
*S*is an uncountable subset of a set*X*endowed with the indiscrete topology, then any point*p*of*X*is a condensation point of*X*as the only open neighborhood of*p*is*X*itself.

## References

- Walter Rudin,
*Principles of Mathematical Analysis*, 3rd Edition, Chapter 2, exercise 27 - John C. Oxtoby,
*Measure and Category*, 2nd Edition (1980), - Lynn Steen and J. Arthur Seebach, Jr.,
*Counterexamples in Topology*, 2nd Edition, pg. 4