The Cauchy pseudocompletion of a pseudometric space
Content created by malarbol and Louis Wasserman.
Created on 2025-10-26.
Last modified on 2025-12-25.
{-# OPTIONS --lossy-unification #-} module metric-spaces.cauchy-pseudocompletion-of-pseudometric-spaces where
Imports
open import elementary-number-theory.addition-positive-rational-numbers open import elementary-number-theory.positive-rational-numbers open import elementary-number-theory.strict-inequality-positive-rational-numbers open import elementary-number-theory.strict-inequality-rational-numbers open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.binary-relations open import foundation.dependent-pair-types open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.transport-along-identifications open import foundation.universe-levels open import metric-spaces.cauchy-approximations-pseudometric-spaces open import metric-spaces.isometries-pseudometric-spaces open import metric-spaces.limits-of-cauchy-approximations-pseudometric-spaces open import metric-spaces.pseudometric-spaces open import metric-spaces.rational-neighborhood-relations open import metric-spaces.short-functions-pseudometric-spaces open import metric-spaces.similarity-of-elements-pseudometric-spaces
Idea
The
Cauchy pseudocompletion¶
of a pseudometric space M is the
pseudometric space of
Cauchy approximations
in M where two Cauchy approximations x and y are in a
d-neighborhood of one
another if for all δ ε : ℚ⁺, x δ and y ε are in a
(δ + ε + d)-neighborhood of one another in M.
Any Cauchy approximation in the Cauchy pseudocompletion has a limit.
Definition
The Cauchy pseudocompletion neighborhood relation on the type of Cauchy approximations in a pseudometric space
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) where neighborhood-prop-cauchy-pseudocompletion-Pseudometric-Space : Rational-Neighborhood-Relation l2 (cauchy-approximation-Pseudometric-Space M) neighborhood-prop-cauchy-pseudocompletion-Pseudometric-Space d x y = Π-Prop ( ℚ⁺) ( λ δ → Π-Prop ( ℚ⁺) ( λ ε → neighborhood-prop-Pseudometric-Space ( M) ( δ +ℚ⁺ ε +ℚ⁺ d) ( map-cauchy-approximation-Pseudometric-Space M x δ) ( map-cauchy-approximation-Pseudometric-Space M y ε))) neighborhood-cauchy-pseudocompletion-Pseudometric-Space : ℚ⁺ → Relation l2 (cauchy-approximation-Pseudometric-Space M) neighborhood-cauchy-pseudocompletion-Pseudometric-Space d x y = type-Prop ( neighborhood-prop-cauchy-pseudocompletion-Pseudometric-Space ( d) ( x) ( y))
Properties
The neighborhood relation is reflexive
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) where abstract is-reflexive-neighborhood-cauchy-pseudocompletion-Pseudometric-Space : (d : ℚ⁺) (x : cauchy-approximation-Pseudometric-Space M) → neighborhood-cauchy-pseudocompletion-Pseudometric-Space M d x x is-reflexive-neighborhood-cauchy-pseudocompletion-Pseudometric-Space d x δ ε = monotonic-neighborhood-Pseudometric-Space M ( map-cauchy-approximation-Pseudometric-Space M x δ) ( map-cauchy-approximation-Pseudometric-Space M x ε) ( δ +ℚ⁺ ε) ( δ +ℚ⁺ ε +ℚ⁺ d) ( le-right-add-rational-ℚ⁺ _ d) ( is-cauchy-approximation-map-cauchy-approximation-Pseudometric-Space ( M) ( x) ( δ) ( ε))
The neighborhood relation is symmetric
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) where abstract is-symmetric-neighborhood-cauchy-pseudocompletion-Pseudometric-Space : (d : ℚ⁺) (x y : cauchy-approximation-Pseudometric-Space M) → neighborhood-cauchy-pseudocompletion-Pseudometric-Space M d x y → neighborhood-cauchy-pseudocompletion-Pseudometric-Space M d y x is-symmetric-neighborhood-cauchy-pseudocompletion-Pseudometric-Space d x y Ndxy δ ε = let xε = map-cauchy-approximation-Pseudometric-Space M x ε yδ = map-cauchy-approximation-Pseudometric-Space M y δ in tr ( λ d' → neighborhood-Pseudometric-Space M d' yδ xε) ( ap (_+ℚ⁺ d) (commutative-add-ℚ⁺ ε δ)) ( symmetric-neighborhood-Pseudometric-Space M _ xε yδ (Ndxy ε δ))
The neighborhood relation is triangular
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) where abstract is-triangular-neighborhood-cauchy-pseudocompletion-Pseudometric-Space : (x y z : cauchy-approximation-Pseudometric-Space M) → (dxy dyz : ℚ⁺) → neighborhood-cauchy-pseudocompletion-Pseudometric-Space M dyz y z → neighborhood-cauchy-pseudocompletion-Pseudometric-Space M dxy x y → neighborhood-cauchy-pseudocompletion-Pseudometric-Space ( M) ( dxy +ℚ⁺ dyz) ( x) ( z) is-triangular-neighborhood-cauchy-pseudocompletion-Pseudometric-Space x y z dxy dyz Ndyz Ndxy δ ε = let xδ = map-cauchy-approximation-Pseudometric-Space M x δ zε = map-cauchy-approximation-Pseudometric-Space M z ε in saturated-neighborhood-Pseudometric-Space ( M) ( δ +ℚ⁺ ε +ℚ⁺ (dxy +ℚ⁺ dyz)) ( xδ) ( zε) ( λ θ → let (θ₂ , θ₂+θ₂<θ) = bound-double-le-ℚ⁺ θ (θa , θb , θa+θb=θ₂) = split-ℚ⁺ θ₂ yθa = map-cauchy-approximation-Pseudometric-Space M y θa in monotonic-neighborhood-Pseudometric-Space ( M) ( xδ) ( zε) ( (δ +ℚ⁺ θa +ℚ⁺ dxy +ℚ⁺ θb) +ℚ⁺ (θa +ℚ⁺ ε +ℚ⁺ dyz +ℚ⁺ θb)) ( δ +ℚ⁺ ε +ℚ⁺ (dxy +ℚ⁺ dyz) +ℚ⁺ θ) ( tr ( λ q → le-ℚ⁺ q (δ +ℚ⁺ ε +ℚ⁺ (dxy +ℚ⁺ dyz) +ℚ⁺ θ)) ( equational-reasoning ((δ +ℚ⁺ ε) +ℚ⁺ (dxy +ℚ⁺ dyz)) +ℚ⁺ (θ₂ +ℚ⁺ θ₂) ＝ ((δ +ℚ⁺ dxy) +ℚ⁺ (ε +ℚ⁺ dyz)) +ℚ⁺ ((θa +ℚ⁺ θb) +ℚ⁺ (θa +ℚ⁺ θb)) by ap-add-ℚ⁺ ( interchange-law-add-add-ℚ⁺ δ ε dxy dyz) ( inv (ap-add-ℚ⁺ θa+θb=θ₂ θa+θb=θ₂)) ＝ ((δ +ℚ⁺ dxy) +ℚ⁺ (θa +ℚ⁺ θb)) +ℚ⁺ ((ε +ℚ⁺ dyz) +ℚ⁺ (θa +ℚ⁺ θb)) by interchange-law-add-add-ℚ⁺ _ _ _ _ ＝ ((δ +ℚ⁺ θa) +ℚ⁺ (dxy +ℚ⁺ θb)) +ℚ⁺ ((ε +ℚ⁺ θa) +ℚ⁺ (dyz +ℚ⁺ θb)) by ap-add-ℚ⁺ ( interchange-law-add-add-ℚ⁺ _ _ _ _) ( interchange-law-add-add-ℚ⁺ _ _ _ _) ＝ (δ +ℚ⁺ θa +ℚ⁺ dxy +ℚ⁺ θb) +ℚ⁺ ((θa +ℚ⁺ ε) +ℚ⁺ (dyz +ℚ⁺ θb)) by ap-add-ℚ⁺ ( inv (associative-add-ℚ⁺ _ _ _)) ( ap-add-ℚ⁺ (commutative-add-ℚ⁺ _ _) refl) ＝ (δ +ℚ⁺ θa +ℚ⁺ dxy +ℚ⁺ θb) +ℚ⁺ (θa +ℚ⁺ ε +ℚ⁺ dyz +ℚ⁺ θb) by ap-add-ℚ⁺ refl (inv (associative-add-ℚ⁺ _ _ _))) ( preserves-le-right-add-ℚ ( rational-ℚ⁺ (δ +ℚ⁺ ε +ℚ⁺ (dxy +ℚ⁺ dyz))) ( rational-ℚ⁺ (θ₂ +ℚ⁺ θ₂)) ( rational-ℚ⁺ θ) ( θ₂+θ₂<θ))) ( triangular-neighborhood-Pseudometric-Space M xδ yθa zε ( δ +ℚ⁺ θa +ℚ⁺ dxy +ℚ⁺ θb) ( θa +ℚ⁺ ε +ℚ⁺ dyz +ℚ⁺ θb) ( monotonic-neighborhood-Pseudometric-Space M yθa zε ( θa +ℚ⁺ ε +ℚ⁺ dyz) ( θa +ℚ⁺ ε +ℚ⁺ dyz +ℚ⁺ θb) ( le-left-add-ℚ⁺ (θa +ℚ⁺ ε +ℚ⁺ dyz) θb) ( Ndyz θa ε)) ( monotonic-neighborhood-Pseudometric-Space M xδ yθa ( δ +ℚ⁺ θa +ℚ⁺ dxy) ( δ +ℚ⁺ θa +ℚ⁺ dxy +ℚ⁺ θb) ( le-left-add-ℚ⁺ (δ +ℚ⁺ θa +ℚ⁺ dxy) θb) ( Ndxy δ θa))))
The neighborhood relation is saturated
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) where abstract is-saturated-neighborhood-cauchy-pseudocompletion-Pseudometric-Space : ( ε : ℚ⁺) (x y : cauchy-approximation-Pseudometric-Space M) → ( (δ : ℚ⁺) → neighborhood-cauchy-pseudocompletion-Pseudometric-Space ( M) ( ε +ℚ⁺ δ) ( x) ( y)) → neighborhood-cauchy-pseudocompletion-Pseudometric-Space M ε x y is-saturated-neighborhood-cauchy-pseudocompletion-Pseudometric-Space d x y H δ ε = let xδ = map-cauchy-approximation-Pseudometric-Space M x δ yε = map-cauchy-approximation-Pseudometric-Space M y ε in saturated-neighborhood-Pseudometric-Space M ( δ +ℚ⁺ ε +ℚ⁺ d) ( xδ) ( yε) ( λ θ → tr ( λ η → neighborhood-Pseudometric-Space M η xδ yε) ( inv (associative-add-ℚ⁺ _ _ _)) ( H θ δ ε))
The Cauchy pseudocompletion of a pseudometric space
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) where cauchy-pseudocompletion-Pseudometric-Space : Pseudometric-Space (l1 ⊔ l2) l2 cauchy-pseudocompletion-Pseudometric-Space = ( cauchy-approximation-Pseudometric-Space M , neighborhood-prop-cauchy-pseudocompletion-Pseudometric-Space M , is-reflexive-neighborhood-cauchy-pseudocompletion-Pseudometric-Space M , is-symmetric-neighborhood-cauchy-pseudocompletion-Pseudometric-Space M , is-triangular-neighborhood-cauchy-pseudocompletion-Pseudometric-Space M , is-saturated-neighborhood-cauchy-pseudocompletion-Pseudometric-Space M)
The isometry from a pseudometric space to its Cauchy pseudocompletion
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) where map-cauchy-pseudocompletion-Pseudometric-Space : type-Pseudometric-Space M → cauchy-approximation-Pseudometric-Space M map-cauchy-pseudocompletion-Pseudometric-Space = const-cauchy-approximation-Pseudometric-Space M abstract preserves-neighborhoods-map-cauchy-pseudocompletion-Pseudometric-Space : (d : ℚ⁺) (x y : type-Pseudometric-Space M) → neighborhood-Pseudometric-Space M d x y → neighborhood-cauchy-pseudocompletion-Pseudometric-Space ( M) ( d) ( map-cauchy-pseudocompletion-Pseudometric-Space x) ( map-cauchy-pseudocompletion-Pseudometric-Space y) preserves-neighborhoods-map-cauchy-pseudocompletion-Pseudometric-Space d x y Nxy δ ε = monotonic-neighborhood-Pseudometric-Space M x y d (δ +ℚ⁺ ε +ℚ⁺ d) ( le-right-add-ℚ⁺ (δ +ℚ⁺ ε) d) ( Nxy) reflects-neighborhoods-map-cauchy-pseudocompletion-Pseudometric-Space : (d : ℚ⁺) (x y : type-Pseudometric-Space M) → neighborhood-cauchy-pseudocompletion-Pseudometric-Space ( M) ( d) ( map-cauchy-pseudocompletion-Pseudometric-Space x) ( map-cauchy-pseudocompletion-Pseudometric-Space y) → neighborhood-Pseudometric-Space M d x y reflects-neighborhoods-map-cauchy-pseudocompletion-Pseudometric-Space d x y Nxy = saturated-neighborhood-Pseudometric-Space M d x y ( λ δ → let (δ₁ , δ₂ , δ₁+δ₂=δ) = split-ℚ⁺ δ in tr ( λ ε → neighborhood-Pseudometric-Space M ε x y) ( commutative-add-ℚ⁺ _ _ ∙ ap (d +ℚ⁺_) δ₁+δ₂=δ) ( Nxy δ₁ δ₂)) is-isometry-map-cauchy-pseudocompletion-Pseudometric-Space : is-isometry-Pseudometric-Space ( M) ( cauchy-pseudocompletion-Pseudometric-Space M) ( map-cauchy-pseudocompletion-Pseudometric-Space) is-isometry-map-cauchy-pseudocompletion-Pseudometric-Space d x y = ( ( preserves-neighborhoods-map-cauchy-pseudocompletion-Pseudometric-Space ( d) ( x) ( y)) , (reflects-neighborhoods-map-cauchy-pseudocompletion-Pseudometric-Space ( d) ( x) ( y))) isometry-cauchy-pseudocompletion-Pseudometric-Space : isometry-Pseudometric-Space ( M) ( cauchy-pseudocompletion-Pseudometric-Space M) isometry-cauchy-pseudocompletion-Pseudometric-Space = ( map-cauchy-pseudocompletion-Pseudometric-Space , is-isometry-map-cauchy-pseudocompletion-Pseudometric-Space) short-map-cauchy-pseudocompletion-Pseudometric-Space : short-function-Pseudometric-Space ( M) ( cauchy-pseudocompletion-Pseudometric-Space M) short-map-cauchy-pseudocompletion-Pseudometric-Space = short-isometry-Pseudometric-Space ( M) ( cauchy-pseudocompletion-Pseudometric-Space M) ( isometry-cauchy-pseudocompletion-Pseudometric-Space)
Convergent Cauchy approximations are similar to constant Cauchy approximations in the Cauchy pseudocompletion
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) (u : cauchy-approximation-Pseudometric-Space M) (x : type-Pseudometric-Space M) where abstract sim-const-is-limit-cauchy-approximation-Pseudometric-Space : is-limit-cauchy-approximation-Pseudometric-Space M u x → sim-Pseudometric-Space ( cauchy-pseudocompletion-Pseudometric-Space M) ( u) ( const-cauchy-approximation-Pseudometric-Space M x) sim-const-is-limit-cauchy-approximation-Pseudometric-Space H d α β = monotonic-neighborhood-Pseudometric-Space ( M) ( map-cauchy-approximation-Pseudometric-Space M u α) ( x) ( α +ℚ⁺ β) ( α +ℚ⁺ β +ℚ⁺ d) ( le-left-add-ℚ⁺ (α +ℚ⁺ β) d) ( H α β) is-limit-sim-const-cauchy-approximation-Pseudometric-Space : sim-Pseudometric-Space ( cauchy-pseudocompletion-Pseudometric-Space M) ( u) ( const-cauchy-approximation-Pseudometric-Space M x) → is-limit-cauchy-approximation-Pseudometric-Space M u x is-limit-sim-const-cauchy-approximation-Pseudometric-Space H α β = saturated-neighborhood-Pseudometric-Space ( M) ( α +ℚ⁺ β) ( map-cauchy-approximation-Pseudometric-Space M u α) ( x) ( λ d → H d α β)
Similarity in the Cauchy pseudocompletion preserves and reflects limits
module _ {l1 l2 : Level} (M : Pseudometric-Space l1 l2) (u v : cauchy-approximation-Pseudometric-Space M) (x : type-Pseudometric-Space M) where has-same-limit-sim-cauchy-approximation-Pseudometric-Space : sim-Pseudometric-Space ( cauchy-pseudocompletion-Pseudometric-Space M) ( u) ( v) → is-limit-cauchy-approximation-Pseudometric-Space M u x → is-limit-cauchy-approximation-Pseudometric-Space M v x has-same-limit-sim-cauchy-approximation-Pseudometric-Space u~v lim-u = is-limit-sim-const-cauchy-approximation-Pseudometric-Space ( M) ( v) ( x) ( transitive-sim-Pseudometric-Space ( cauchy-pseudocompletion-Pseudometric-Space M) ( v) ( u) ( const-cauchy-approximation-Pseudometric-Space M x) ( sim-const-is-limit-cauchy-approximation-Pseudometric-Space ( M) ( u) ( x) ( lim-u)) ( inv-sim-Pseudometric-Space ( cauchy-pseudocompletion-Pseudometric-Space M) ( u~v))) sim-has-same-limit-cauchy-approximation-Pseudometric-Space : is-limit-cauchy-approximation-Pseudometric-Space M u x → is-limit-cauchy-approximation-Pseudometric-Space M v x → sim-Pseudometric-Space ( cauchy-pseudocompletion-Pseudometric-Space M) ( u) ( v) sim-has-same-limit-cauchy-approximation-Pseudometric-Space lim-u lim-v = transitive-sim-Pseudometric-Space ( cauchy-pseudocompletion-Pseudometric-Space M) ( u) ( const-cauchy-approximation-Pseudometric-Space M x) ( v) ( inv-sim-Pseudometric-Space ( cauchy-pseudocompletion-Pseudometric-Space M) ( sim-const-is-limit-cauchy-approximation-Pseudometric-Space ( M) ( v) ( x) ( lim-v))) ( sim-const-is-limit-cauchy-approximation-Pseudometric-Space M u x lim-u)
Recent changes
- 2025-12-25. malarbol. Refactor function actions on cauchy approximations (#1738).
- 2025-11-06. malarbol and Louis Wasserman. Metric quotients of pseudometric spaces (#1622).
- 2025-10-26. malarbol and Louis Wasserman. Cauchy pseudocompletions of (pseudo)metric spaces (#1619).