Vol. 74, No. 1, 1978

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The boundedness of the Cantor-Bendixson order of some analytic sets

R. Daniel Mauldin

Vol. 74 (1978), No. 1, 167–177

Let X and Y be complete separable metric spaces (Polish spaces). If E is a subset of X × Y , and x X, then by the x-section of E, Ex, is meant E ({xY ). By Ps(E) is

PS(E ) = {x : Ex is scattered}.

In this paper the following uniform boundedness principle for the Cantor-Bendixson order of analytic sets will be demonstrated.

Theorem L. Let W be an analytic subset of X × Y and let M be an analytic subset of X such that M PS(W). Then there is a countable ordinal α such that the α-th Cantor-Bendixson derived set of Ex is empty, for each x in M.

Mathematical Subject Classification 2000
Primary: 54H05
Received: 15 March 1976
Published: 1 January 1978
R. Daniel Mauldin
Department of Mathematics
University of North Texas
Denton TX 76203-1430
United States