arXiv subjects
Publications and source records attributed to Andrea Medini.
We construct a homogeneous subspace of $2^ω$ whose complement is dense in $2^ω$ and rigid. Using the same method, assuming Martin's Axiom, we also construct a countable dense homogeneous subspace of $2^ω$ whose complement is dense in $2^ω$ and rigid.
All spaces are assumed to be separable and metrizable. Our main result is that the statement "For every space $X$, every closed subset of $X$ has the perfect set property if and only if every analytic subset of $X$ has the perfect set property" is equivalent to $\mathfrak{b}>ω_1$ (hence, in particular, it is independent of $\mathsf{ZFC}$). This, together with a theorem of Solecki and an example of Miller, will allow us to determine the status of the statement "For every space $X$, if every $\mathbfΓ$ subset of $X$ has the perfect set property then every $\mathbfΓ'$ subset of $X$ has the perfect set property" as $\mathbfΓ,\mathbfΓ'$ range over all pointclasses of complexity at most analytic or coanalytic. Along the way, we define and investigate a property of independent interest. We will say that a subset $W$ of $2^ω$ has the Grinzing property if it is uncountable and for every uncountable $Y\subseteq W$ there exists an uncountable collection consisting of uncountable subsets of $Y$ with pairwise disjoint closures in $2^ω$. The following theorems hold. (1) There exists a subset of $2^ω$ with the Grinzing property. (2) Assume $\mathsf{MA}+\neg\mathsf{CH}$. Then $2^ω$ has the Grinzing property. (3) Assume $\mathsf{CH}$. Then $2^ω$ does not have the Grinzing property. The first result was obtained by Miller using a theorem of Todorčević, and is needed in the proof of our main result.
Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we construct a zero-dimensional separable metrizable space $X$ such that $X$ is countable dense homogeneous while $X^2$ is not. It follows from results of Hrušák and Zamora Avilés that such a space $X$ cannot be Borel. Furthermore, $X$ can be made homogeneous and completely Baire as well.
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 $2^ω$. (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.
Building on work of Terada, we prove that h-homogeneity is productive in the class of zero-dimensional spaces. Then, by generalizing a result of Motorov, we show that for every non-empty zero-dimensional space $X$ there exists a non-empty zero-dimensional space $Y$ such that $X\times Y$ is h-homogeneous. Also, we simultaneously generalize results of Motorov and Terada by showing that if $X$ is a space such that the isolated points are dense then $X^κ$ is h-homogeneous for every infinite cardinal $κ$. Finally, we show that a question of Terada (whether $X^ω$ is h-homogeneous for every zero-dimensional first-countable $X$) is equivalent to a question of Motorov (whether such an infinite power is always divisible by 2) and give some partial answers.
We construct a family of Hausdorff spaces such that every finite product of spaces in the family (possibly with repetitions) is CLP-compact, while the product of all spaces in the family is non-CLP-compact. Our example will yield a single Hausdorff space $X$ such that every finite power of $X$ is CLP-compact, while no infinite power of $X$ is CLP-compact. This answers a question of Steprāns and Šostak.
Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $ω$ up to homeomorphism. Here, we identify ultrafilters with subpaces of $2^ω$ in the obvious way. Using the same methods, still under Martin's Axiom for countable posets, we will construct a non-principal ultrafilter $\UU\subseteq 2^ω$ such that $\UU^ω$ is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial results about the relation of such topological properties with the combinatorial property of being a $\mathrm{P}$-point.