Cauchy approximations in metric spaces

Content created by malarbol, Fredrik Bakke and Louis Wasserman.

Created on 2024-09-28.
Last modified on 2025-08-18.

module metric-spaces.cauchy-approximations-metric-spaces where
Imports 
open import elementary-number-theory.positive-rational-numbers

open import foundation.constant-maps
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.propositions
open import foundation.subtypes
open import foundation.universe-levels

open import metric-spaces.cauchy-approximations-pseudometric-spaces
open import metric-spaces.metric-spaces
open import metric-spaces.short-functions-metric-spaces

Idea

A Cauchy approximation in a metric space A is a Cauchy approximation in its underlying pseudometric space: a map f from ℚ⁺ to the carrier type of A such that for all positive rationals ε and δ, f ε and f δ are in a (ε + δ)-neighborhood, i.e., the distance between f ε and f δ is bounded by ε + δ.

Definitions

Cauchy approximations in metric spaces

module _
  {l1 l2 : Level} (A : Metric-Space l1 l2)
  where

  is-cauchy-approximation-prop-Metric-Space :
    (ℚ⁺ → type-Metric-Space A) → Prop l2
  is-cauchy-approximation-prop-Metric-Space =
    is-cauchy-approximation-prop-Pseudometric-Space
      ( pseudometric-Metric-Space A)

  is-cauchy-approximation-Metric-Space :
    (ℚ⁺ → type-Metric-Space A) → UU l2
  is-cauchy-approximation-Metric-Space =
    type-Prop ∘ is-cauchy-approximation-prop-Metric-Space

  cauchy-approximation-Metric-Space : UU (l1 ⊔ l2)
  cauchy-approximation-Metric-Space =
    type-subtype is-cauchy-approximation-prop-Metric-Space

module _
  {l1 l2 : Level} (A : Metric-Space l1 l2)
  (f : cauchy-approximation-Metric-Space A)
  where

  map-cauchy-approximation-Metric-Space :
    ℚ⁺ → type-Metric-Space A
  map-cauchy-approximation-Metric-Space =
    map-cauchy-approximation-Pseudometric-Space
      ( pseudometric-Metric-Space A)
      ( f)

  is-cauchy-approximation-map-cauchy-approximation-Metric-Space :
    (ε δ : ℚ⁺) →
    neighborhood-Metric-Space
      ( A)
      ( ε +ℚ⁺ δ)
      ( map-cauchy-approximation-Metric-Space ε)
      ( map-cauchy-approximation-Metric-Space δ)
  is-cauchy-approximation-map-cauchy-approximation-Metric-Space =
    is-cauchy-approximation-map-cauchy-approximation-Pseudometric-Space
      ( pseudometric-Metric-Space A)
      ( f)

Properties

Constant maps in metric spaces are Cauchy approximations

module _
  {l1 l2 : Level} (A : Metric-Space l1 l2)
  (x : type-Metric-Space A)
  where

  const-cauchy-approximation-Metric-Space :
    cauchy-approximation-Metric-Space A
  const-cauchy-approximation-Metric-Space =
    const-cauchy-approximation-Pseudometric-Space
      ( pseudometric-Metric-Space A)
      ( x)

The action of short maps on Cauchy approximations

module _
  {l1 l2 l1' l2' : Level}
  (A : Metric-Space l1 l2) (B : Metric-Space l1' l2')
  (f : short-function-Metric-Space A B)
  where

  map-short-function-cauchy-approximation-Metric-Space :
    cauchy-approximation-Metric-Space A →
    cauchy-approximation-Metric-Space B
  map-short-function-cauchy-approximation-Metric-Space =
    map-short-function-cauchy-approximation-Pseudometric-Space
      ( pseudometric-Metric-Space A)
      ( pseudometric-Metric-Space B)
      ( f)

module _
  {l1 l2 : Level}
  (A : Metric-Space l1 l2)
  where

  eq-id-map-short-function-cauchy-approximation-Metric-Space :
    map-short-function-cauchy-approximation-Metric-Space
      ( A)
      ( A)
      ( short-id-Metric-Space A) ＝
    id
  eq-id-map-short-function-cauchy-approximation-Metric-Space = refl

module _
  {l1a l2a l1b l2b l1c l2c : Level}
  (A : Metric-Space l1a l2a)
  (B : Metric-Space l1b l2b)
  (C : Metric-Space l1c l2c)
  (g : short-function-Metric-Space B C)
  (f : short-function-Metric-Space A B)
  where

  eq-comp-map-short-function-cauchy-approximation-Metric-Space :
    ( map-short-function-cauchy-approximation-Metric-Space B C g ∘
      map-short-function-cauchy-approximation-Metric-Space A B f) ＝
    ( map-short-function-cauchy-approximation-Metric-Space A C
      (comp-short-function-Metric-Space A B C g f))
  eq-comp-map-short-function-cauchy-approximation-Metric-Space = refl

Homotopic Cauchy approximations are equal

module _
  { l1 l2 : Level} (A : Metric-Space l1 l2)
  { f g : cauchy-approximation-Metric-Space A}
  ( f~g :
    map-cauchy-approximation-Metric-Space A f ~
    map-cauchy-approximation-Metric-Space A g)
  where

  eq-htpy-cauchy-approximation-Metric-Space : f ＝ g
  eq-htpy-cauchy-approximation-Metric-Space =
    eq-htpy-cauchy-approximation-Pseudometric-Space
      ( pseudometric-Metric-Space A)
      ( f~g)

References

Our definition of Cauchy approximation follows Definition 4.5.5 of [Boo20] and Definition 11.2.10 of [UF13].

[Boo20]
Auke Bart Booij. Analysis in univalent type theory. PhD thesis, University of Birmingham, 2020. URL: https://etheses.bham.ac.uk/id/eprint/10411/7/Booij20PhD.pdf.
[UF13]
The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study, 2013. URL: https://homotopytypetheory.org/book/, arXiv:1308.0729.

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