{"id":"palomar/PALOMAR-2026-10-28-000032","url":"https://github.com/Arthur742Ramos/Math-Formalization/tree/eee893e9a0359f39eb6c8ff3fcf50244b743fa9f/topology/e0-nonsmoothness","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-28-000032-v1.json","repo":"https://github.com/Arthur742Ramos/Math-Formalization","tags":["math.LO","math.GN","msc:03E15","msc:54H05","msc:54E52"],"score":0,"title":"Generic ergodicity and non-smoothness of E₀","trust":"high","domain":"math.LO","source":"palomar","authors":["Arthur Freitas Ramos","David Barros Hulak","Ruy Jose Guerra Barretto de Queiroz"],"reviews":0,"subject":"math.LO","summary":"Eventual equality on the actual Cantor product has a BP invariant-set zero-one law. Every invariant map with BP open preimages into a second countable T0 target has a comeagre constant fiber. Consequently no such BP map, and no Borel map with explicit Borel sigma-algebras, reduces eventual equality to equality.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-28","reactions":{},"formalized":true,"formalizes":[{"title":"Baire Category","identifier":"https://homepages.math.uic.edu/~rosendal/WebpagesMathCourses/MATH511-notes/DST%20notes%20-%20Kuratowski-Ulam08.pdf","relationship":"formalizes"},{"title":"The Smooth Ideal","identifier":"https://doi.org/10.1112/plms/pdv069","relationship":"formalizes"}]}