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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 $\sigma$-homogeneous with analytic witnesses, (2) Every finite-dimensional analytic space is $\sigma$-homogeneous with pairwise disjoint $\mathbf{\Delta}^1_2$ witnesses. Furthermore, the complexity of the witnesses is optimal in both of the above results. This completes the picture regarding $\sigma$-homogeneity in the finite-dimensional realm. It is an open problem whether every analytic space is $\sigma$-homogeneous. We also investigate finite unions of homogeneous spaces., Comment: 10 pages. arXiv admin note: text overlap with arXiv:2107.07747

Let $\mathcal{L}$ be a first-order two-sorted language and consider a class of $\mathcal{L}$-structures of the form $\langle M, X \rangle$ where $M$ varies among structures of the first sort, while $X$ is fixed in the second sort, and it is assumed to be a compact Hausdorff space. When $X$ is a compact subset of the real line, one way to treat classes of this kind model-theoretically is via continuous-valued logic, as in [Ben Yaacov-Berenstein-Henson-Usvyatsov 2010]. Prior to that, Henson and Iovino proposed an approach based on the notion of positive formulas [Henson-Iovino 2002]. Their work is tailored to the model theory of Banach spaces. Here we show that a similar approach is possible for a more general class of models. We introduce suitable versions of elementarity, compactness, saturation, quantifier elimination and other basic tools, and we develop basic model theory.

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., Comment: 16 pages

In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard $\kappa$-Borel spaces for $\kappa$ an uncountable (regular) cardinal satisfying $\kappa^{<\kappa} = \kappa$. As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work from [Coskey-Schlicht 2016, Galeotti 2019, Luecke-Schlicht 2015], and answer some questions contained therein., Comment: 43 pages, accepted for publication on the Journal of the London Mathematical Society

Recently, Solecki introduced the notion of Ramsey monoid to produce a common generalization to theorems such as Hindman's theorem, Carlson's theorem, and Gowers' FIN$_k$ theorem. He proved that an entire class of finite monoids is Ramsey. Here we improve this result, enlarging this class and finding a simple algebraic characterization of finite Ramsey monoids. We extend in a similar way a result of Solecki regarding a second class of monoids connected to the Furstenberg-Katznelson Ramsey Theorem. The results obtained suggest a possible connection with Sch\"utzenberger's theorem and finite automata theory., Comment: The structure of the paper has been strongly reworked, and some sections have been removed, accordingly to the referee's suggestions. Added new results on Solecki's generalization of Furstenberg and Katznelson's theorem. Added references for aperiodic monoids

We present a characterization of spaces of strictly decreasing functions on trees in terms of bisequentiality. This characterization answers Questions 6.1 and 6.2 of "A filter on a collection of finite sets and Eberlein compacta" by T. Cie\'sla. Moreover we study the relation between these spaces and the classes of Corson, Eberlein and uniform Eberlein compacta., Comment: 9 pages