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= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

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


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