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Cantor-Bendixson theorem

CUP theorem
Type of Math Object: 
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Mathematics Subject Classification

03E15 no label found54H05 no label found


Looking at the proof, I think this theorem holds for any second countable T_1 topological space. Are there any other generalizations of this theorem?

Also, the proof that uses the Cantor-Bendixson rank of X requires an auxilliary result: a strictly decreasing chain of closed sets is at most countable. Kuratowski (Topology), attributes this result to Baire. Does anyone know if this theorem has a commonly accepted name?

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