Between Polish and completely Baire

TL;DR

This paper explores the relationships between several topological properties of separable metrizable spaces, including Polishness, Miller property, Cantor-Bendixson property, and complete Baire, providing new counterexamples and analyzing their definability.

Contribution

It establishes the implications among these properties, provides counterexamples within ZFC and under various set-theoretic assumptions, and investigates how continuum size influences definability.

Findings

Implications (1)→(2)→(3)→(4) hold universally.

Counterexamples show the non-reversibility of implications in ZFC and other models.

Every uncountable completely Baire space has size continuum.

Abstract

All spaces are assumed to be separable and metrizable. Consider the following properties of a space $X$. (1) $X$ is Polish. (2) For every countable crowded $Q\subseteq X$ there exists a crowded $Q'\subseteq Q$ with compact closure. (3) Every closed subspace of $X$ is either scattered or it contains a homeomorphic copy of $2^\omega$. (4) Every closed subspace of $X$ is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications $(1)\rightarrow (2)\rightarrow (3)\rightarrow (4)$ hold for every space $X$. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if $X$ is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a $\mathsf{ZFC}$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication $(i)\leftarrow (i+1)$ for $i=1,2,3$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum.