The metric space of Cauchy approximations in a saturated complete metric space
Content created by malarbol.
Created on 2025-05-01.
Last modified on 2025-05-01.
module metric-spaces.metric-space-of-cauchy-approximations-saturated-complete-metric-spaces where
Imports
open import elementary-number-theory.positive-rational-numbers open import foundation.dependent-pair-types open import foundation.function-types open import foundation.universe-levels open import metric-spaces.cauchy-approximations-metric-spaces open import metric-spaces.complete-metric-spaces open import metric-spaces.convergent-cauchy-approximations-metric-spaces open import metric-spaces.dependent-products-metric-spaces open import metric-spaces.metric-space-of-cauchy-approximations-complete-metric-spaces open import metric-spaces.metric-space-of-cauchy-approximations-metric-spaces open import metric-spaces.metric-space-of-convergent-cauchy-approximations-metric-spaces open import metric-spaces.metric-spaces open import metric-spaces.saturated-complete-metric-spaces open import metric-spaces.saturated-metric-spaces open import metric-spaces.short-functions-metric-spaces open import metric-spaces.subspaces-metric-spaces
Idea
The type of
Cauchy approximations in
a
saturated complete metric space
A
inherits the
metric structure
of the Cauchy approximations in the underlying metric space; this is the
saturated complete metric space of Cauchy approximations¶
in a saturated complete metric space.
All Cauchy approximations in a saturated complete metric space are convergent and the map from a Cauchy approximation to its limit is short.
Definitions
The metric space of Cauchy approximations in a saturated complete metric space
module _ {l1 l2 : Level} (A : Saturated-Complete-Metric-Space l1 l2) where metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space : Metric-Space (l1 ⊔ l2) l2 metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space = metric-space-of-cauchy-approximations-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A)
Properties
The metric space of Cauchy approximations in a saturated complete metric space is saturated
module _ {l1 l2 : Level} (A : Saturated-Complete-Metric-Space l1 l2) where is-saturated-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space : is-saturated-Metric-Space (metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A) is-saturated-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space = is-saturated-metric-space-of-cauchy-approximations-is-saturated-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A) ( is-saturated-metric-space-Saturated-Complete-Metric-Space A)
The map from a Cauchy approximation in a saturated complete metric space to its limit is short
module _ {l1 l2 : Level} (A : Saturated-Complete-Metric-Space l1 l2) where short-limit-cauchy-approximation-Saturated-Complete-Metric-Space : short-function-Metric-Space ( metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A) ( metric-space-Saturated-Complete-Metric-Space A) short-limit-cauchy-approximation-Saturated-Complete-Metric-Space = comp-short-function-Metric-Space ( metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A) ( metric-space-of-convergent-cauchy-approximations-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A)) ( metric-space-Saturated-Complete-Metric-Space A) ( short-limit-convergent-cauchy-approximation-is-saturated-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A) ( is-saturated-metric-space-Saturated-Complete-Metric-Space A)) ( short-convergent-cauchy-approximation-Complete-Metric-Space ( complete-metric-space-Saturated-Complete-Metric-Space A)) is-short-limit-cauchy-approximation-Saturated-Complete-Metric-Space : is-short-function-Metric-Space ( metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A) ( metric-space-Saturated-Complete-Metric-Space A) ( limit-cauchy-approximation-Saturated-Complete-Metric-Space A) is-short-limit-cauchy-approximation-Saturated-Complete-Metric-Space = is-short-map-short-function-Metric-Space ( metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A) ( metric-space-Saturated-Complete-Metric-Space A) ( short-limit-cauchy-approximation-Saturated-Complete-Metric-Space)
The metric space of Cauchy approximations in a saturated complete metric space is complete
Given a Cauchy approximation of Cauchy approximations U : ℚ⁺ → ℚ⁺ → A
in a
saturated complete metric space A
, we construct its limit as follows:
- for any
η : ℚ⁺
, the partial applicationε ↦ U ε η
is a Cauchy approximation inA
; - since
A
is complete, it converges to somelim-η : A
; - since
A
is saturated, the assignmentη ↦ lim-η
is a Cauchy approximation inA
; - by construction it’s a limit of
U
in the space of Cauchy approximations.
module _ {l1 l2 : Level} (A : Saturated-Complete-Metric-Space l1 l2) (U : cauchy-approximation-Metric-Space ( metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A)) where map-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space : ℚ⁺ → type-Saturated-Complete-Metric-Space A map-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space η = limit-cauchy-approximation-Complete-Metric-Space ( complete-metric-space-Saturated-Complete-Metric-Space A) ( swap-cauchy-approximation-cauchy-approximations-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A) ( U) ( η)) is-cauchy-map-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space : is-cauchy-approximation-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A) ( map-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space) is-cauchy-map-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space ε δ = is-short-limit-cauchy-approximation-Saturated-Complete-Metric-Space ( A) ( ε +ℚ⁺ δ) ( swap-cauchy-approximation-cauchy-approximations-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A) ( U) ( ε)) ( swap-cauchy-approximation-cauchy-approximations-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A) ( U) ( δ)) ( λ η → is-cauchy-approximation-map-cauchy-approximation-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A) ( map-cauchy-approximation-Metric-Space ( metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space ( A)) ( U) ( η)) ( ε) ( δ)) lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space : cauchy-approximation-Metric-Space (metric-space-Saturated-Complete-Metric-Space A) lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space = map-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space , is-cauchy-map-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space is-limit-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space : is-limit-cauchy-approximation-Metric-Space ( metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space ( A)) ( U) ( lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space) is-limit-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space ε δ η = is-limit-limit-cauchy-approximation-Complete-Metric-Space ( complete-metric-space-Saturated-Complete-Metric-Space A) ( swap-cauchy-approximation-cauchy-approximations-Metric-Space ( metric-space-Saturated-Complete-Metric-Space A) ( U) ( η)) ( ε) ( δ)
module _ {l1 l2 : Level} (A : Saturated-Complete-Metric-Space l1 l2) where is-complete-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space : is-complete-Metric-Space ( metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A) is-complete-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space U = ( lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space ( A) ( U)) , ( is-limit-lim-cauchy-approximation-cauchy-approximations-Saturated-Complete-Metric-Space ( A) ( U))
The saturated complete metric space of Cauchy approximations in a saturated complete metric space
module _ {l1 l2 : Level} (A : Saturated-Complete-Metric-Space l1 l2) where saturated-complete-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space : Saturated-Complete-Metric-Space (l1 ⊔ l2) l2 pr1 saturated-complete-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space = metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A pr2 saturated-complete-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space = ( is-complete-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A) , ( is-saturated-metric-space-of-cauchy-approximations-Saturated-Complete-Metric-Space A)