Topology of higher Baire spaces

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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.