Limit point of a set

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A point each neighbourhood of which contains at least one point of the given set different from it. The point and set considered are regarded as belonging to a topological space. A set containing all its limit points is called closed. The set of all limit points of a set $M$ is called the derived set, and is denoted by $M'$. If the topological space $X$ satisfies the first separation axiom (for any two points $x$ and $y$ in it there is a neighbourhood $U(x)$ of $x$ not containing $y$), then every neighbourhood of a limit point of a set $M\subset X$ contains infinitely many points of this set and the derived set $M'$ is closed. Every proximate point of a set $M$ is either a limit point or an isolated point of it.


[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2] F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))


A limit point of a set is usually called an accumulation point of that set. See also (the editorial comments to) Condensation point.


[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Limit point of a set. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article