M-separable spaces of functions are productive in the Miller model

Distinguishing perfect set properties in separable metrizable spaces

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 perfec…

零维 $\sigma$-齐性空间

Zero-dimensional $σ$-homogeneous spaces

本文在不同集合论公理下研究零维空间的 $\sigma$-齐性，给出正反结论，引入遗传刚性概念并重证相关结果。

SPM Bulletin 19

Contents of the issue: Selection Principles and special sets of reals: Open problems Winning the pressing down game but not Banach Mazur Ramsey classes of topological and metric spaces More on partitioning triples of countable ordinals Countable compact Hausdorff spaces need not be metrizable in ZF Every topological group is a group retract of a minimal group The complexity of classifying separable Banach spaces up to isomorphism Reals n-generic relative to some perfect tree Nagata's conjecture and countably compactifications in generic extensions A Class of Groups in Which All Unconditionally…

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…