Theorems · Theorem · general topology
isCompact_singleton
∀ {X : Type u} [inst : TopologicalSpace X] {x : X}, IsCompact {x}- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 32 results in Mathlib
- Foundations
- Depth 61 from the axioms · uses propext, Quot.sound
- Assumes
- TopologicalSpace
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 · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Filterproof · cited by 8,121
- nhdsproof · cited by 5,554
- LE.le.transproof · cited by 3,151
- IsCompactstatement · cited by 1,282
- Filter.NeBotproof · cited by 853
- Filter.principalproof · cited by 740
- pure_le_nhdsproof · cited by 39
- Filter.principal_singletonproof · cited by 31
- ClusterPt.of_le_nhds'proof · cited by 3
Cited by32
Results whose statement or proof uses this declaration.
- Set.Finite.isCompactproof · cited by 10
- isProperMap_iff_isClosedMap_and_compact_fibersproof · cited by 5
- IsCompact.insertproof · cited by 3
- RCLike.geometric_hahn_banach_closed_pointproof · cited by 3
- NumberField.mixedEmbedding.fundamentalCone.isCompact_compactSetproof · cited by 3
- isProperMap_iff_isCompact_preimageproof · cited by 3
- intervalIntegral.intervalIntegrable_cpowproof · cited by 2
- geometric_hahn_banach_closed_pointproof · cited by 2
- ArzelaAscoli.isCompact_of_equicontinuousproof · cited by 2
- IsProperMap.restrictPreimageproof · cited by 2
- Set.Subsingleton.isCompactproof · cited by 2
- IsLocalHomeomorph.exists_lift_nhdsproof · cited by 1