Theorems · Theorem · general topology
Metric.isCompact_iff_isClosed_bounded
- 1000+ list: Heine–Borel theorem
∀ {α : Type u_3} {s : Set α} [inst : MetricSpace α] [ProperSpace α], IsCompact s ↔ IsClosed s ∧ Bornology.IsBounded sThe Heine–Borel theorem: In a proper metric space, a set is compact if and only if it is closed and bounded.
- Defined in
- Mathlib.Topology.MetricSpace.Bounded
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 152 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MetricSpaceProperSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- MetricSpacestatement and proof · cited by 1,684
- IsClosedstatement and proof · cited by 1,639
- IsCompactstatement and proof · cited by 1,282
- Bornology.IsBoundedstatement and proof · cited by 293
- ProperSpacestatement and proof · cited by 190
- IsCompact.isClosedproof · cited by 77
- IsCompact.isBoundedproof · cited by 29
- Metric.isCompact_of_isClosed_isBoundedproof · cited by 8
Cited by3
Results whose statement or proof uses this declaration.
- ModularGroup.isCompact_truncatedFundamentalDomainproof · cited by 1
- AntilipschitzWith.properSpaceproof · cited by 1
- NumberField.mixedEmbedding.convexBodySum_compactproof · cited by 1