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Alexandrov–Urysohn characterization of Baire space

Arthur Freitas Ramos, David Barros Hulak, Ruy Jose Guerra Barretto de Queiroz

Palomar Registrypalomar/PALOMAR-2026-10-22-000042Oct 22, 2026math.GNscore 0

A topological space is homeomorphic to Baire space, the countable product of discrete natural numbers, if and only if it is nonempty, Polish, has a basis of clopen sets, and every compact subset has empty interior. The construction partitions each nonempty clopen set into countably infinitely many distinct nonempty clopen children with arbitrarily small diameter. A fully branching shrinking scheme gives a continuous open bijective limit map and therefore a homeomorphism. The converse uses finite coordinate images of compact Baire subsets. This formalizes a classical theorem of Alexandrov and…

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