Theorems · Theorem · general topology
IsClosed.isCompact
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X} [CompactSpace X], IsClosed s → IsCompact s- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 43 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceCompactSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- IsClosedstatement and proof · cited by 1,639
- IsCompactstatement · cited by 1,282
- CompactSpacestatement and proof · cited by 593
- Set.subset_univproof · cited by 228
- IsCompact.of_isClosed_subsetproof · cited by 67
- isCompact_univproof · cited by 53
Cited by43
Results whose statement or proof uses this declaration.
- Continuous.isClosedMapproof · cited by 9
- AlgebraicGeometry.exists_map_eq_topproof · cited by 5
- TopCat.nonempty_limitCone_of_compact_t2_cofiltered_systemproof · cited by 2
- connectedComponent_eq_iInter_isClopenproof · cited by 2
- PrimeSpectrum.exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopenproof · cited by 2
- ContinuousMap.idealOfSet_ofIdeal_eq_closureproof · cited by 2
- CompHausLike.isClosedMapproof · cited by 2
- MDifferentiable.isLocallyConstantproof · cited by 2
- exists_idempotent_of_compact_t2_of_continuous_add_leftproof · cited by 2
- exists_idempotent_of_compact_t2_of_continuous_mul_leftproof · cited by 2
- IsLocalHomeomorph.exists_lift_nhdsproof · cited by 1
- IsTopologicalAddGroup.exist_add_closure_nhdsproof · cited by 1