{"id":"palomar/PALOMAR-2026-10-22-000042","url":"https://github.com/Arthur742Ramos/Math-Formalization/tree/b77053ba8669ad046ec3c0d54d83a79724935a9f/topology/baire-space-characterization","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-22-000042-v1.json","repo":"https://github.com/Arthur742Ramos/Math-Formalization","tags":["math.GN","math.LO","msc:54E50","msc:54F65","msc:03E15","general-topology","baire-space","descriptive-set-theory","topology-characterization"],"score":0,"title":"Alexandrov–Urysohn characterization of Baire space","trust":"high","domain":"math.GN","source":"palomar","authors":["Arthur Freitas Ramos","David Barros Hulak","Ruy Jose Guerra Barretto de Queiroz"],"reviews":0,"subject":"math.GN","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-22","reactions":{},"formalized":true,"formalizes":[{"title":"Introduction to descriptive set theory","identifier":"https://www.math.mcgill.ca/atserunyan/Teaching_notes/dst_lectures.pdf","relationship":"formalizes"},{"title":"Descriptive set theory problem set","identifier":"https://homepages.math.uic.edu/~cdean8/docs/dst_problems.pdf","relationship":"formalizes"}]}