# Condensation point of a set

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].)