Condensation point

In mathematics, a condensation point p of a subset S of a topological space is any point p such that every neighborhood of p contains uncountably many points of S. Thus "condensation point" is synonymous with " $\aleph _{1}$ -accumulation point".^[1]^[2]

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 neighborhood of p is X itself.

References

^ Efimov, B. A. (2001) [1994], "Condensation point of a set", Encyclopedia of Mathematics, EMS Press
^ Lynn Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, 2nd Edition, pg. 5

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