A primitive associated with the Cantor–Bendixson derivative on Polish spaces

Given a perfect Polish space X, a compact subset K ⊆ X and a countable ordinal α < ω1, we show that there exists a compact subset K^b ⊆ X such that K^b (α⁾ = K, where K^b (α⁾ denotes the α-th Cantor–Bendixson derivative of K^b . In other words, every compact subset of a perfect Polish space admits an α-primitive with respect to the Cantor–Bendixson derivative. This extends to perfect Polish spaces a result previously known for countable compact subsets of the real line. The proof proceeds in three steps: first, we construct primitives for singletons; then, for countable compact subsets; and finally, for arbitrary compact subsets, using separability of Polish spaces.