Topological aspects of higher Baire space

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Many interesting properties of the real numbers can be studied by focusing attention on the irrational numbers, and there is a convenient equivalence between the irrationals and the set of all functions from natural numbers to natural numbers. A recent major focus of research in set theory has been in studying a generalisation of this, with some larger set taking the place of the natural numbers. In this research project the student will study topological aspects of this generalisation - that is, understanding it in terms of a notion of "closeness". As well as the intrinsic interest in understanding this natural mathematical structure, the research may lead to applications in other areas of mathematics such as model theory, where constraining attention to the natural numbers is a significant restriction. From there results and general understanding may flow on to applications in unexpected areas.