Countable dense homogeneity of definable spaces

Countable dense homogeneity and $λ$-sets

We show that all sufficiently nice $λ$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $κ\le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $κ$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $κ$ is equivalent to the existence of a $λ$-set of size $κ$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $ω_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ metric s…

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…

波兰空间的大乘积中的可数稠密齐性

Countable dense homogeneity in large products of Polish spaces

该研究统一处理波兰空间乘积的可数稠密齐性问题，重点关注不可数乘积，给出少于$\mathfrak{p}$个波兰空间乘积满足可数稠密齐性的条件，得到新应用并证明结果最优。

Products and h-homogeneity

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-dime…

$σ$-homogeneity of Borel sets

We give an affirmative answer to the following question: Is any Borel subset of a Cantor set $\textbf{ C}$ a sum of a countable number of pairwise disjoint $h$-homogeneous subspaces that are closed in $X$? It follows that every Borel set $X \subset \textbf{ R}^n$ can be partitioned into countably many $h$-homogeneous subspaces that are $G_δ$-sets in $X$.

On homogeneous locally conical spaces

The main result of this article is: THEOREM. Every homogeneous locally conical connected separable metric space that is not a $1$-manifold is strongly $n$-homogeneous for each $n \geq 2$ and countable dense homogeneous. Furthermore, countable dense homogeneity can be proven without assuming the space is connected. This theorem has the following two consequences. COROLLARY 1. If $X$ is a homogeneous compact suspension, then $X$ is an absolute suspension (i.e., for any two distinct points $p$ and $q$ of $X$, there is a homeomorphism from $X$ to a suspension that maps $p$ and $q$ to the suspensio…

The topology of ultrafilters as subspaces of $2^ω$

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 resu…

在可数余数的奇异基数上的一般描述集合论

Generalized Descriptive Set Theory at Singular Cardinals of Countable Cofinality

本文在可数余数的奇异基数上发展了描述集合论，研究了λ-波兰空间、λ-博雷尔空间及相关集合的性质，并推广了经典完美集性质。

Between Polish and completely Baire

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 re…