arXiv subjects

Andrea Medini

Publications and source records attributed to Andrea Medini.

At least 19 recordsLinked to original sources

arXiv2026

Non-meager $\mathsf{P}$-filters, Miller-measurability, and a question of Hru\v{s}\'{a}k

Given a cardinal $\kappa$ and filters $\mathcal{F}_\alpha$ on $\omega$ for $\alpha\in\kappa$, we will show that if $\prod_{\alpha\in\kappa}\mathcal{F}_\alpha$ is countable dense homogeneous then $\kappa<\mathfrak{p}$ and each $\mathcal{F}_\alpha$ is a non-meager $\mathsf{P}$-filter. This partially answers a question of Michael Hru\v{s}\'{a}k. Along the way, we will show that the product of fewer than $\mathfrak{p}$ non-meager $\mathsf{P}$-filters has the Miller property. We will also describe explicitly the connection between Miller-measurability and the Miller property. As a corollary, we will see that the intersection of fewer than $\mathsf{add}(m^0)$ non-meager $\mathsf{P}$-filters is a non-meager $\mathsf{P}$-filter, where $m^0$ denotes the ideal of Miller-null sets. We will conclude by investigating the preservation of the Miller property under intersections and products.

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arXiv2025

Countable dense homogeneity in large products of Polish spaces

We give a unified treatment of the countable dense homogeneity of products of Polish spaces, with a focus on uncountable products. Our main result states that a product of fewer than $\mathfrak{p}$ Polish spaces is countable dense homogeneous if the following conditions hold: (1) Each factor is strongly locally homogeneous, (2) Each factor is strongly $n$-homogeneous for every $n\in\omega$, (3) Every countable subset of the product can be brought in general position. For example, using the above theorem, one can show that $2^\kappa$, $\omega^\kappa$, $\mathbb{R}^\kappa$ and $[0,1]^\kappa$ are countable dense homogeneous for every infinite $\kappa <\mathfrak{p}$ (these results are due to Stepr\={a}ns and Zhou, except for the one concerning $\omega^\kappa$). In fact, as a new application, we will show that every product of fewer than $\mathfrak{p}$ connected manifolds with boundary is countable dense homogeneous, provided that none or infinitely many of the boundaries are non-empty. This generalizes a result of Yang. Along the way, we will discuss and employ several results concerning the general position of countable sets. Finally, we will show that our main result and its corollaries are optimal.

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arXiv2025

Countable dense homogeneity and topological groups

Building on results of Medvedev, we construct a $\mathsf{ZFC}$ example of a non-Polish topological group that is countable dense homogeneous. Our example is a dense subgroup of $\mathbb{Z}^\omega$ of size $\mathfrak{b}$ that is a $\lambda$-set. We also conjecture that every countable dense homogenous Baire topological group with no isolated points contains a copy of the Cantor set, and give a proof in a very special case.

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arXiv2025

On the infinite powers of large zero-dimensional metrizable spaces

We show that $X^\lambda$ is strongly homogeneous whenever $X$ is a non-separable zero-dimensional metrizable space and $\lambda$ is an infinite cardinal. This partially answers a question of Terada, and improves a previous result of the author. Along the way, we show that every non-compact weight-homogeneous metrizable space with a $\pi$-base consisting of clopen sets can be partitioned into $\kappa$ many clopen sets, where $\kappa$ is the weight of $X$. This improves a result of van Engelen.

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arXiv2024

Every finite-dimensional analytic space is $σ$-homogeneous

All spaces are assumed to be separable and metrizable. Building on work of van Engelen, Harrington, Michalewski and Ostrovsky, we obtain the following results: (1) Every finite-dimensional analytic space is $σ$-homogeneous with analytic witnesses, (2) Every finite-dimensional analytic space is $σ$-homogeneous with pairwise disjoint $\mathbfΔ^1_2$ witnesses. Furthermore, the complexity of the witnesses is optimal in both of the above results. This completes the picture regarding $σ$-homogeneity in the finite-dimensional realm. It is an open problem whether every analytic space is $σ$-homogeneous. We also investigate finite unions of homogeneous spaces.

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arXiv2023

A complete classification of the zero-dimensional homogeneous spaces under determinacy

All spaces are assumed to be separable and metrizable. We give a complete classification of the zero-dimensional homogeneous spaces, under the Axiom of Determinacy. This classification is expressed in terms of topological complexity (in the sense of Wadge theory) and Baire category. In the same spirit, we also give a complete classification of the filters on $\omega$ up to homeomorphism. As byproducts, we obtain purely topological characterizations of the semifilters and filters on $\omega$. The Borel versions of these results are in almost all cases due to Fons van Engelen. Along the way, we obtain Wadge-theoretic results of independent interest, especially regarding closure properties.

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arXiv2023

Countable spaces, realcompactness, and the pseudointersection number

All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows that, given a cardinal $\kappa$, the following conditions are equivalent: $(1)$ There exists a countable crowded space $X$ such that $\mathsf{Exp}(X)=\kappa$, $(2)$ $\mathfrak{p}\leq\kappa\leq\mathfrak{c}$. In fact, in the case $\mathfrak{d}\leq\kappa\leq\mathfrak{c}$, every countable dense subspace of $2^\kappa$ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $\kappa$ has pseudocharacter at most $\kappa$ in any compactification. This will allow us to calculate $\mathsf{Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space.

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arXiv2023

Zero-dimensional $σ$-homogeneous spaces

All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is $σ$-homogeneous. Inspired by this theorem, we obtain the following results: assuming $\mathsf{AD}$, every zero-dimensional space is $σ$-homogeneous; assuming $\mathsf{AC}$, there exists a zero-dimensional space that is not $σ$-homogeneous; assuming $\mathsf{V=L}$, there exists a coanalytic zero-dimensional space that is not $σ$-homogeneous. Along the way, we introduce two notions of hereditary rigidity, and give alternative proofs of results of van Engelen, Miller and Steel. It is an open problem whether every analytic zero-dimensional space is $σ$-homogeneous.

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arXiv2022

Constructing Wadge classes

We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $ω_1$ (that is, the ones that are closed under Borel preimages) and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep (namely, that every non-selfdual Wadge class can be expressed as the result of a Hausdorff operation applied to the open sets). The exposition is self-contained, except for facts from classical descriptive set theory.

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arXiv2022

On the scope of the Effros theorem

All spaces (and groups) are assumed to be separable and metrizable. Jan van Mill showed that every analytic group $G$ is Effros (that is, every continuous transitive action of $G$ on a non-meager space is micro-transitive). We complete the picture by obtaining the following results: under $\mathsf{AC}$, there exists a non-Effros group; under $\mathsf{AD}$, every group is Effros; under $\mathsf{V=L}$, there exists a coanalytic non-Effros group. The above counterexamples will be graphs of discontinuous homomorphisms.

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arXiv2020

Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (that is, all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

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arXiv2017

Productively Lindelöf spaces of countable tightness

Michael asked whether every productively Lindelöf space is powerfully Lindelöf. Building of work of Alster and De la Vega, assuming the Continuum Hypothesis, we show that every productively Lindelöf space of countable tightness is powerfully Lindelöf. This strengthens a result of Tall and Tsaban. The same methods also yield new proofs of results of Arkhangel'skii and Buzyakova. Furthermore, assuming the Continuum Hypothesis, we show that a productively Lindelöf space $X$ is powerfully Lindelöf if every open cover of $X^ω$ admits a point-continuum refinement consisting of basic open sets. This strengthens a result of Burton and Tall. Finally, we show that separation axioms are not relevant to Michael's question: if there exists a counterexample (possibly not even $\mathsf{T}_0$), then there exists a regular (actually, zero-dimensional) counterexample.

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arXiv2017

Non-meager free sets and independent families

Our main result is that, given a collection $\mathcal{R}$ of meager relations on a Polish space $X$ such that $|\mathcal{R}|\leqω$, there exists a dense Baire subspace $F$ of $X$ (equivalently, a nowhere meager subset $F$ of $X$) such that $F$ is $R$-free for every $R\in\mathcal{R}$. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on $ω$, and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting "$|\mathcal{R}|\leqω$" with "$|\mathcal{R}|<\mathfrak{c}$" and "Baire" with "completely Baire".

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arXiv2017

Infinite powers and Cohen reals

We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of $Z^ω$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of $ω_1$ Cohen reals, viewed as a subspace of $2^ω$.

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arXiv2016

On Borel semifilters

Building on work of van Engelen and van Mill, we show that a zero-dimensional Borel space is homeomorphic to a semifilter if and only if it is homogeneous and not locally compact. Under $\mathbfΣ^1_1$-Determinacy, this result extends to all analytic and coanalytic spaces.

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arXiv2016

Every filter is homeomorphic to its square

We show that every filter $\mathcal{F}$ on $ω$, viewed as a subspace of $2^ω$, is homeomorphic to $\mathcal{F}^2$. This generalizes a theorem of van Engelen, who proved that this holds for Borel filters.

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arXiv2015

Countable dense homogeneity in powers of zero-dimensional definable spaces

We show that, for a coanalytic subspace $X$ of $2^ω$, the countable dense homogeneity of $X^ω$ is equivalent to $X$ being Polish. This strengthens a result of Hrušák and Zamora Avilés. Then, inspired by results of Hernández-Gutiérrez, Hrušák and van Mill, using a technique of Medvedev, we construct a non-Polish subspace $X$ of $2^ω$ such that $X^ω$ is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer to a question of Hrušák and Zamora Avilés. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a question of Medini and Milovich. Finally, we show that if every countable subset of a zero-dimensional separable metrizable space $X$ is included in a Polish subspace of $X$ then $X^ω$ is countable dense homogeneous.

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arXiv2014

Seven characterizations of non-meager P-filters

We give several topological/combinatorial conditions that, for a filter on $ω$, are equivalent to being a non-meager $\mathsf{P}$-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager $\mathsf{P}$-filter. Here, we identify a filter with a subspace of $2^ω$ through characteristic functions. Along the way, we generalize to non-meager $\mathsf{P}$-filters a result of Miller about $\mathsf{P}$-points, and we employ and give a new proof of results of Marciszewski. We also employ a theorem of Hernández-Gutiérrez and Hrušák, and answer two questions that they posed. Our result also resolves several issues raised by Medini and Milovich, and proves false one "theorem" of theirs. Furthermore, we show that the statement "Every non-meager filter contains a non-meager $\mathsf{P}$-subfilter" is independent of $\mathsf{ZFC}$ (more precisely, it is a consequence of $\mathfrak{u}<\mathfrak{g}$ and its negation is a consequence of $\Diamond$). It follows from results of Hrušák and van Mill that, under $\mathfrak{u}<\mathfrak{g}$, a filter has less than $\mathfrak{c}$ types of countable dense subsets if and only if it is a non-meager $\mathsf{P}$-filter. In particular, under $\mathfrak{u}<\mathfrak{g}$, there exists an ultrafilter with $\mathfrak{c}$ types of countable dense subsets. We also show that such an ultrafilter exists under $\mathsf{MA(countable)}$.

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