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 "${\displaystyle \aleph _{1}}$-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.