Countable dense homogeneity in large products of Polish spaces

Abstract

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.