Theorems · Theorem · general topology
IsCompact.isClosed
∀ {X : Type u_1} [inst : TopologicalSpace X] [T2Space X] {s : Set X}, IsCompact s → IsClosed sIn a T2Space, every compact set is closed.
- Defined in
- Mathlib.Topology.Separation.Hausdorff
- Cited by
- 77 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceT2Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- TopologicalSpacestatement and proof · cited by 24,529
- Filterproof · cited by 8,121
- nhdsproof · cited by 5,554
- IsClosedstatement · cited by 1,639
- T2Spacestatement and proof · cited by 1,351
- IsCompactstatement and proof · cited by 1,282
- Filter.NeBotproof · cited by 853
- Filter.principalproof · cited by 740
- ClusterPtproof · cited by 138
- ClusterPt.monoproof · cited by 30
- Set.mem_of_eq_of_memproof · cited by 8
Cited by77
Results whose statement or proof uses this declaration.
- IsCompact.measurableSetproof · cited by 25
- Continuous.isClosedMapproof · cited by 9
- Function.locallyFinsuppWithin.finiteSupportproof · cited by 9
- HasCompactSupport.tsupport_extend_zero_subsetproof · cited by 5
- HasCompactMulSupport.mulTSupport_extend_one_subsetproof · cited by 4
- Topology.RelCWComplex.isClosed_closedCellproof · cited by 4
- rieszContentAux_image_nonemptyproof · cited by 3
- LightProfinite.epi_iff_surjectiveproof · cited by 3
- ae_eq_zero_of_integral_contMDiff_smul_eq_zeroproof · cited by 3
- isProperMap_iff_isCompact_preimageproof · cited by 3
- Metric.isCompact_iff_isClosed_boundedproof · cited by 3
- isCompact_setOfPred_finiteMeasure_le_of_isCompactproof · cited by 2