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.”]

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