Condensation point

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 Thus, "condensation point" is synonymous with "$\aleph _{1}$-accumulation point".

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