Theorems · Theorem · general topology
Metric.IsCover.isCompact
∀ {X : Type u_1} [inst : MetricSpace X] [ProperSpace X] {ε : NNReal} {s N : Set X},
Metric.IsCover ε s N → IsClosed s → IsCompact N → IsCompact sA closed set in a proper metric space which admits a compact cover is compact.
- Defined in
- Mathlib.Topology.MetricSpace.Cover
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 158 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.
Cites11
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
- NNRealstatement and proof · cited by 4,310
- MetricSpacestatement and proof · cited by 1,684
- IsClosedstatement and proof · cited by 1,639
- IsCompactstatement and proof · cited by 1,282
- ProperSpacestatement and proof · cited by 190
- IsCompact.isClosedproof · cited by 77
- IsCompact.of_isClosed_subsetproof · cited by 67
- Metric.IsCoverstatement and proof · cited by 51
- IsCompact.cthickeningproof · cited by 4
- Metric.IsCover.subset_cthickeningproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Metric.IsCover.isCompact_closureproof · cited by 1