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 $A$ is a subset of a topological space $X$ and $x$ is a point of $X$, then $x$ is an accumulation point of $A$ if and only if every neighbourhood of $x$ intersects $A\setminus\{x\}$. It is a condensation point of $A$ if and only if every neighbourhood of it contains uncountably many points of $A$.

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

In the case of a $T_1$-space $X$ the notions of a limit point of a set $A\subset X$ and an accumulation point of $A$ 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