Condensation Point

A condensation point of a set, within a given topological space, is a point with the property that every open neighborhood of it contains uncountably many points of the set, a stronger requirement than merely being a limit point, which only requires infinitely many nearby points. The notion sharpens the ordinary idea of a limit point specifically to separate countable accumulation from the much denser accumulation associated with uncountable sets, and it plays a central role in one of the classical results of descriptive set theory, the Cantor-Bendixson theorem, which shows that every closed set of real numbers can be split uniquely into a perfect set, consisting entirely of its own condensation points, together with a countable remainder. The theorem and the underlying condensation-point construction, developed out of work by Georg Cantor and put into its modern form by Ivar Bendixson in the 1880s, gave one of the first demonstrations that an arbitrary closed set of real numbers has a highly constrained structure rather than an arbitrary one. Condensation points remain a standard tool in general topology and descriptive set theory wherever a set needs to be decomposed according to how densely its points accumulate.