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Formally continuous functions on Baire space
Author(s) -
Kawai Tatsuji
Publication year - 2018
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201700015
Subject(s) - mathematics , baire space , baire category theorem , uniform continuity , baire measure , countable set , pure mathematics , complete metric space , continuous function (set theory) , morphism , discrete mathematics , equivalence (formal languages) , function (biology) , metric space , evolutionary biology , biology
A function from Baire space N N to the natural numbers N is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working in Bishop constructive mathematics: one is a function induced by a Brouwer‐operation (i.e., inductively defined neighbourhood function); the other is a function uniformly continuous near every compact image. We show that formal continuity is equivalent to the former while it is strictly stronger than the latter. The equivalence of formally continuous functions and those induced by Brouwer‐operations requires Countable Choice.