Condensation point of a set

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in a Euclidean space $E^n$

A point of $E^n$ such that every neighbourhood of it contains uncountable many points of the set. The set of condensation points of a set is always closed; if it is non-empty, it is perfect and has the cardinality of the continuum. The concept of a condensation point can be generalized to arbitrary topological spaces.


The generalization to arbitrary spaces is direct: A point $x$ a condensation point (of a set $M$) in a topological space if (the intersection of $M$ with) every neighbourhood of is an uncountable set. (See also [a1].)


[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
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Condensation point of a set. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article