Theorems · Theorem · general topology
Metric.isCompact_of_isClosed_isBounded
∀ {α : Type u} {s : Set α} [inst : PseudoMetricSpace α] [ProperSpace α],
IsClosed s → Bornology.IsBounded s → IsCompact sThe Heine–Borel theorem: In a proper space, a closed bounded set is compact.
- Defined in
- Mathlib.Topology.MetricSpace.Bounded
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpaceProperSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- Realproof · cited by 25,697
- Set.Nonemptyproof · cited by 2,627
- IsClosedstatement and proof · cited by 1,639
- PseudoMetricSpacestatement and proof · cited by 1,550
- IsCompactstatement · cited by 1,282
- Metric.closedBallproof · cited by 704
- Bornology.IsBoundedstatement and proof · cited by 293
- Set.eq_empty_or_nonemptyproof · cited by 248
- ProperSpacestatement and proof · cited by 190
- IsCompact.of_isClosed_subsetproof · cited by 67
- ProperSpace.isCompact_closedBallproof · cited by 40
Cited by8
Results whose statement or proof uses this declaration.
- Bornology.IsBounded.measure_lt_topproof · cited by 8
- Bornology.IsBounded.isCompact_closureproof · cited by 6
- spectrum.isCompactproof · cited by 5
- IsCompact.cthickeningproof · cited by 4
- exists_contDiff_tsupport_subsetproof · cited by 3
- Metric.isCompact_iff_isClosed_boundedproof · cited by 3
- Continuous.exists_forall_le_of_isBoundedproof · cited by 2
- Metric.compactSpace_iff_isBounded_univproof · cited by 1