Condensation point

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See Condensation point of a set; Limit point of a set; Accumulation point.


The three notions mentioned above should be clearly distinguished. If is a subset of a topological space and is a point of , then is an accumulation point of if and only if every neighbourhood of intersects . It is a condensation point of if and only if every neighbourhood of it contains uncountably many points of .

The term limit point is slightly ambiguous. One might call a limit point of if every neighbourhood of contains infinitely many points of , but this is not standard. Sometimes one calls a limit point of a net (cf. Net (of sets in a topological space)) if some subset of this net converges to . However, most people call a cluster point in this case.

In the case of a -space the notions of a limit point of a set and an accumulation point of coincide, and one uses "accumulation point" .

How to Cite This Entry:
Condensation point. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article