  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.
