Derived set

The collection $M'$ of all limit points of a set $M$ in a topological space (cf. Limit point of a set). A set $M$ that coincides with its derived set is called perfect.

Comments

This process can be iterated.

In general one defines, for an ordinal number $\alpha$, the $\alpha$-th derived set of $X$, $X^{(\alpha)}$, as follows: $X^{(0)} = X$, $X^{(\alpha+1)}$ is the derived set of $X^{(\alpha)}$, and if $\lambda$ is a limit ordinal then $X^{(\lambda)} = \bigcap_{\alpha < \lambda} X^{(\alpha)}$.

One then shows that there is a first ordinal number $\alpha = \alpha_X$ such that $X^{(\alpha+1)} = X^{(\alpha)}$. If $X^{(\alpha)} = \emptyset$, then $X$ is called scattered; if $X^{(\alpha)} \neq \emptyset$, then $X^{(\alpha)}$ is called the perfect kernel of $X$.

In this way one can prove the Cantor–Bendixson theorem: If $X$ is a subspace of the real line, then $X = C \cup P$, with $C$ a countable set, $P$ a perfect set and $C \cap P = \emptyset$.

For this reason $\alpha_X$ is sometimes called the Cantor–Bendixson height of $X$. Perfect spaces are sometimes called dense-in-itself.