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

Andrés Merino; Sebastián Heredia Freire

doi:10.56754/0719-0646.2802.227

Authors

Andrés Merino Facultad de Ciencias Exactas, Naturales y Ambientales, Pontificia Universidad Católica del Ecuador, Quito, Ecuador. https://orcid.org/0000-0002-5404-918X
Sebastián Heredia Freire School of Mathematical and Computational Sciences, Yachay Tech University, San Miguel de Urcuquí, Ecuador. https://orcid.org/0000-0003-0604-1244

DOI:

https://doi.org/10.56754/0719-0646.2802.227

Keywords:

Cantor–Bendixson derivative, descriptive set theory, Polish spaces, primitive

Abstract

Given a perfect Polish space \(X\), a compact subset \(K\subset X\) and a countable ordinal \(\alpha<\omega_1\), we show that there exists a compact subset \(\widehat K\subset X\) such that

\[
\widehat K^{(\alpha)} = K
\]

where \(\widehat K^{(\alpha)}\) denotes the \(\alpha\)-th Cantor–Bendixson derivative of \(\widehat K\). In other words, every compact subset of a perfect Polish space admits an \(\alpha\)-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.

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