Baire space

Content created by Fredrik Bakke, Jonathan Prieto-Cubides, Egbert Rijke, Elif Uskuplu and Victor Blanchi.

Created on 2022-06-30.
Last modified on 2023-06-25.

module set-theory.baire-space where
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.lawveres-fixed-point-theorem
open import foundation.negation
open import foundation.propositional-truncations
open import foundation.universe-levels

open import foundation-core.empty-types
open import foundation-core.identity-types
open import foundation-core.propositions
open import foundation-core.sets

open import set-theory.countable-sets
open import set-theory.uncountable-sets


The Baire space is the type of functions ℕ → ℕ.


baire-space : UU lzero
baire-space =   


The Baire Space is a set

is-set-baire-space : is-set baire-space
is-set-baire-space f g =
    ( λ p q 
      ( inv (is-retraction-eq-htpy p)) 
      ( ( ap
          ( eq-htpy)
            ( eq-htpy
              ( λ n 
                  ( is-set-ℕ (f n) (g n))
                  ( htpy-eq p n)
                  ( htpy-eq q n)))) 
      ( is-retraction-eq-htpy q)))

baire-space-Set : Set lzero
pr1 baire-space-Set = baire-space
pr2 baire-space-Set = is-set-baire-space

The Baire Space is uncountable

is-uncountable-baire-space : is-uncountable baire-space-Set
is-uncountable-baire-space P =
    ( is-directly-countable-is-countable baire-space-Set succ-ℕ P)
    ( empty-Prop)
    ( λ H 
        ( fixed-point-theorem-Lawvere (pr2 H) succ-ℕ)
        ( empty-Prop)
        ( λ F 
          reductio-ad-absurdum (pr2 F) (has-no-fixed-points-succ-ℕ (pr1 F))))

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