## 502 – Cantor-Bendixson derivatives

November 8, 2009

Given a topological space $X$ and a set $B\subseteq X,$ let $B'$ be the set of accumulation points of $B,$ i.e., those points $p$ of $X$ such that any open neighborhood of $p$ meets $B$ in an infinite set.

Suppose that $B$ is closed. Then $B'\subseteq B.$ Define $B^\alpha$ for $B$ closed compact by recursion: $B^0=B,$ $B^{\alpha+1}=(B^\alpha)',$ and $B^\lambda=\bigcap_{\alpha<\lambda}B^\alpha$ for $\lambda$ limit. Note that this is a decreasing sequence, so that if we set $B^\infty=\bigcap_{\alpha\in{\sf ORD}}B^\alpha,$ there must be an $\alpha$ such that $B^\infty=B^\beta$ for all $\beta\ge\alpha.$

[The sets $B^\alpha$ are the Cantor-Bendixson derivatives of $B.$ In general, a derivative operation is a way of associating to sets $B$ some kind of “boundary.”]