Images of metric spaces under short functions

Content created by Louis Wasserman.

Created on 2025-08-31.
Last modified on 2025-08-31.

module metric-spaces.images-short-functions-metric-spaces where
Imports 
open import foundation.dependent-pair-types
open import foundation.images
open import foundation.universe-levels

open import metric-spaces.functions-metric-spaces
open import metric-spaces.images-metric-spaces
open import metric-spaces.metric-spaces
open import metric-spaces.short-functions-metric-spaces

Idea

Given a short function between metric spaces f : X → Y, the unit map of the image of f is short.

Definition

module _
  {l1 l2 l3 l4 : Level} (X : Metric-Space l1 l2) (Y : Metric-Space l3 l4)
  (f : short-function-Metric-Space X Y)
  where

  im-short-function-Metric-Space : Metric-Space (l1 ⊔ l3) l4
  im-short-function-Metric-Space =
    im-Metric-Space X Y (map-short-function-Metric-Space X Y f)

  map-unit-im-short-function-Metric-Space :
    type-function-Metric-Space X im-short-function-Metric-Space
  map-unit-im-short-function-Metric-Space =
    map-unit-im (map-short-function-Metric-Space X Y f)

Properties

The unit map of the image of a short function is short

module _
  {l1 l2 l3 l4 : Level} (X : Metric-Space l1 l2) (Y : Metric-Space l3 l4)
  (f : short-function-Metric-Space X Y)
  where

  is-short-map-unit-im-short-function-Metric-Space :
    is-short-function-Metric-Space
      ( X)
      ( im-short-function-Metric-Space X Y f)
      ( map-unit-im-short-function-Metric-Space X Y f)
  is-short-map-unit-im-short-function-Metric-Space =
    is-short-map-short-function-Metric-Space X Y f

  short-map-unit-im-short-function-Metric-Space :
    short-function-Metric-Space X (im-short-function-Metric-Space X Y f)
  short-map-unit-im-short-function-Metric-Space =
    ( map-unit-im-short-function-Metric-Space X Y f ,
      is-short-map-unit-im-short-function-Metric-Space)

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