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Dense subsets of the real numbers #1750

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Dense subsets of the real numbers
lowasser Dec 11, 2025
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Update src/real-numbers/dense-subsets-real-numbers.lagda.md
lowasser Dec 11, 2025
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18 changes: 18 additions & 0 deletions src/metric-spaces/dense-subsets-metric-spaces.lagda.md
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Expand Up @@ -46,6 +46,24 @@ module _
is-dense-subset-Metric-Space : UU (l1 ⊔ l2 ⊔ l3)
is-dense-subset-Metric-Space = type-Prop is-dense-prop-subset-Metric-Space
dense-subset-Metric-Space :
{l1 l2 : Level} (l3 : Level) (X : Metric-Space l1 l2) → UU (l1 ⊔ l2 ⊔ lsuc l3)
dense-subset-Metric-Space l3 X =
type-subtype (is-dense-prop-subset-Metric-Space {l3 = l3} X)
module _
{l1 l2 l3 : Level}
(X : Metric-Space l1 l2)
(S : dense-subset-Metric-Space l3 X)
where
subset-dense-subset-Metric-Space : subset-Metric-Space l3 X
subset-dense-subset-Metric-Space = pr1 S
is-dense-subset-dense-subset-Metric-Space :
is-dense-subset-Metric-Space X subset-dense-subset-Metric-Space
is-dense-subset-dense-subset-Metric-Space = pr2 S
```

## Properties
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2 changes: 2 additions & 0 deletions src/real-numbers.lagda.md
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Expand Up @@ -24,6 +24,7 @@ open import real-numbers.cauchy-completeness-dedekind-real-numbers public
open import real-numbers.cauchy-sequences-real-numbers public
open import real-numbers.closed-intervals-real-numbers public
open import real-numbers.dedekind-real-numbers public
open import real-numbers.dense-subsets-real-numbers public
open import real-numbers.difference-real-numbers public
open import real-numbers.distance-real-numbers public
open import real-numbers.enclosing-closed-rational-intervals-real-numbers public
Expand Down Expand Up @@ -87,6 +88,7 @@ open import real-numbers.positive-real-numbers public
open import real-numbers.powers-real-numbers public
open import real-numbers.proper-closed-intervals-real-numbers public
open import real-numbers.raising-universe-levels-real-numbers public
open import real-numbers.rational-approximates-of-real-numbers public
open import real-numbers.rational-lower-dedekind-real-numbers public
open import real-numbers.rational-real-numbers public
open import real-numbers.rational-upper-dedekind-real-numbers public
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