Topology of higher Baire spaces

这个项目的目的是评估在多大程度上某些基本结果的真实的线可以概括的背景下，更大的无限集。主要的好处将是一个更深入的理解这两个原始结果的真实的线和强公理的大无限集。在短到可以预见的时间尺度上，它可能会应用于模型理论，但主要是好奇心驱动的“蓝天”研究，旨在更好地了解世界。它属于EPSRC的“逻辑与组合学”研究领域。更详细地说：无理数集，作为实数的子空间，与Baire空间Ω ^Ω同胚，并且实数的许多有趣的拓扑方面，如Baire范畴和勒贝格可测性，都是在该设置下有效地研究的。近年来，很多注意力集中在将Baire空间推广到它的更高类似物kappa^kappa，因为kappa是不可数基数。欧米加情形中的一些事实很容易概括，但其他事实则不然--例如，在卡帕情形中，勒贝格-空集的理想没有被广泛接受的类似物，尽管描述这样一个理想是一个正在进行的研究的主题。该项目将进一步研究这些广义Baire空间的性质。

This project aims to assess the extent to which certain fundamental results about the real line can be generalised in the context of larger infinite sets. The main benefits will be a deeper understanding of both the original results about the real line and strong axioms about large infinite sets. In timescales short enough to be forseeable, it may have applications to model theory, but is primarily curiosity-driven "blue skies" research, undertaken to gain a greater understanding of the world. It fits into the "Logic and Combinatorics" research area of the EPSRC portfolio.In more detail: the set of irrational numbers, as a subspace of the reals, is homeomorphic to the Baire space omega^omega, and many interesting topological aspects of the reals such as Baire category and Lebesgue measurability are just as effectively studied in that setting. In recent years much attention has focused on generalising Baire space to its higher analogues kappa^kappa for kappa an uncountable cardinal. Some facts from the omega case generalise easily, but others do not - for example, there is no widely-accepted analogue for the ideal of Lebesgue-null sets in the kappa case, although characterising such an ideal is a topic of ongoing research. This project would study further the properties of these generalised Baire spaces.