Theorems · Theorem · general topology
Metric.eq_countable_union_of_isBounded_of_isOpen
∀ {α : Type u} [inst : PseudoMetricSpace α] {U : Set α},
IsOpen U → ∃ f, Monotone f ∧ ⋃ i, f i = U ∧ ∀ (i : ℕ), Bornology.IsBounded (f i) ∧ IsOpen (f i)Every open set in a metric space is a countable union of bounded open sets.
- Defined in
- Mathlib.Topology.MetricSpace.Bounded
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Set.Nonemptyproof · cited by 2,627
- Set.iUnionstatement · cited by 2,483
- IsOpenstatement and proof · cited by 2,400
- PseudoMetricSpacestatement and proof · cited by 1,550
- Monotonestatement · cited by 1,397
- Metric.ballproof · cited by 735
- Set.inter_subset_rightproof · cited by 329
- Bornology.IsBoundedstatement · cited by 293
- Set.eq_empty_or_nonemptyproof · cited by 248
- Set.inter_univproof · cited by 198
- Nat.cast_leproof · cited by 159
Cited by1
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.OuterRegular.ext_isOpen_isBoundedproof · cited by 0