Topology of higher Baire spaces

这个项目旨在评估关于实线的某些基本结果可以在更大的无限集的背景下推广的程度。主要的好处是对关于实线的原始结果和关于大无穷集的强公理有了更深的理解。在足够短的可预见的时间尺度上，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。它符合EPSRC投资组合的“逻辑学与组合学”研究领域。更详细地说：无理数的集合，作为实数的子空间，是与贝尔空间同胚的，实数的许多有趣的拓扑方面，如贝尔范畴和勒贝格可测性，都在这种情况下得到了有效的研究。近年来，对于不可数基数kappa，人们将Baire空间推广到它的更高类似物kappa^kappa。omega情况下的一些事实很容易推广，但其他事实则不然——例如，kappa情况下的Lebesgue-null集理想没有被广泛接受的类似物，尽管描述这种理想是一个正在进行的研究课题。本课题将进一步研究这些广义贝尔空间的性质。

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.