The Boundedness of the Cantor-bendixson Order of Some Analytic Sets

A = Γ\β<a {%x is an acculumation point of A and x e A}. Recall that a subset, H, of a Polish space is scattered if, and only if H is a countable Gδ set, or equivalently, there is a countable ordinal 7 such that the 7th Cantor-Bendixson derived set, H\ of H is empty [5]. By the Cantor-Bendixson order of a subset H of a topological space is meant the first ordinal 7 such that H = H. The Cantor-Bendixson order of every subset of a Polish space is necessarily less than ωx [5]. If Ec X x Yand I d , then E will be bounded on M provided there is an ordinal 7, 7 < ωlf such that for each x in Λf, the CantorBendixson order of Ex is < 7̂; otherwise i? will be said to be unbounded on ikf. Let us note that in order to prove Theorem L it suffices to show that if E is an analytic subset of X x Y such that each ce-section of E is scattered then E is bounded on the X projection of E, πx(E). Theorem L has the following corollary: