The first part of these notes give an introduction to the theory of Polish group actions on compact Hausdorff spaces, leading up to a proof of the Kechris-Pestov-Todorcevic correspondence and discussions of properties of universal minimal flows. The second part proveides some background on descriptive set theory and culminates with B. Miller's proof of the $\mathcal{G}_0$-dichotomy theorem due to Kechris, Solecki, and Todorcevic.
Kunen's proof of the non-existence of Reinhardt cardinals opened up the research on very large cardinals, i.e., hypotheses at the limit of inconsistency. One of these large cardinals, I0, proved to have descriptive-set-theoretical characteristics, similar to those implied by the Axiom of Determinacy: if $λ$ witnesses I0, then there is a topology for $V_{λ+1}$ that is completely metrizable and with weight $λ$ (i.e., it is a $λ$-Polish space), and it turns out that all the subsets of $V_{λ+1}$ in $L(V_{λ+1})$ have the $λ$-Perfect Set Property in such topology. In this paper, we find another gene…
Quasi-Polish spaces were introduced by de Brecht as a possibly non-Hausdorff generalization of Polish spaces sharing many of their descriptive set-theoretic properties. We give a self-contained exposition of the basic theory of quasi-Polish spaces, based on their "logical" characterization as $\mathbfΠ^0_2$ subspaces of countable powers of Sierpinski space, with several new proofs emphasizing this point of view as well as making more extensive use of Baire category techniques.
In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups, groups of exponential growth and groups with decidable word problem are $\mathbfΣ^0_2$-complete and that the sets of periodic groups and groups of intermediate growth are $\mathbfΠ^0_2$-complete. We also provide bounds for the descriptive complexity of simplicity, amenability, residually finiteness, Hopficity and co-Hopficity. This paper is intended to serve as…
We introduce a new topological generalization of the $σ$-projective hierarchy, not limited to Polish spaces. Earlier attempts have replaced $^ωω$ by $^κκ$, for $κ$ regular uncountable, or replaced countable by $σ$-discrete. Instead we close the usual $σ$-projective sets under continuous images and perfect preimages together with countable unions. The natural set-theoretic axiom to apply is $σ$-projective determinacy, which follows from large cardinals. Our goal is to generalize the known results for $K$-analytic spaces (continuous images of perfect preimages of $^ωω$) to these more general set…
Borel and Hausdorff Hierarchies in Topological Spaces of Choquet Games and Their Effectivization
在波兰空间中完成的经典描述集合论的哪些部分仍然适用于更一般的拓扑空间，可能是T0或T1，但不适用于T2（即不适用于豪斯多夫）？这个问题已经解决了维克多Selivanov在一系列文件集中的代数域。最近，马修·德·布莱希特（Matthew de Brecht）考虑了拟波兰空间，这是一个包含可数基连续域和波兰空间的框架。在本文中，我们提出了替代统一拓扑空间，我们称之为近似空间。在Choquet博弈中，Nonempty玩家在这些空间上有固定策略。近似空间的一个自然真子类与拟波兰空间类重合。我们研究了近似空间中的Borel和Hausdorff差族，重温了其他拓扑空间所做的工作。我们还考虑这些结果的有效性的问题。
A rank is a notion in descriptive set theory that describes ranks such as the Cantor-Bendixson rank on the set of closed subsets of a Polish space, differentiability ranks on the set of differentiable functions in $C[0,1]$ such as the Kechris-Woodin rank and many other ranks in descriptive set theory and real analysis. The complexity of many natural ranks is $Π^1_1$ or $Σ^1_2$. We propose to understand the least length of ranks on a set as a measure of its complexity. Therefore, the aim is to understand which lengths such ranks may have. The main result determines the suprema of lengths of cou…
