Condensation point

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Definition[edit]

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 "\aleph_1-accumulation point".[citation needed]

Examples[edit]

  • 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[edit]