# Limit point of a set

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.

#### References

 [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))