On the Borel class of the derivea set operator
نویسندگان
چکیده
KURATOWSKI showed that the derived set operator Z), acting on the space 2^ of closed subsets of a metric space X, is a Borel map of class exactly two and posed the problem of determining the precise classes of the higher order derivatives 0. We show that the exact classes of the higher derivatives D" are unbounded in ©i. In particular, we show that D* is not of class a and that, for limit ordinals A D^ is of Borel class exactly A + 1. The proof involves the construction of a sub-lattice ̂ of the space of closed subsets of2 on which (1) both the union and intersection maps are continuous lattice homomorphisms, (2) Z) is a lattice homomorphism, and (3) the derived set order map is a lattice homomorphism, and (3) the derived set order map is a lattice homomorphism into (O
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