Formally continuous functions on Baire space

10/24/2017

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by   Tatsuji Kawai, et al.

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A function from Baire space to the natural numbers 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.