Theorems · Theorem · general topology
IsCompact.of_isClosed_subset
∀ {X : Type u} [inst : TopologicalSpace X] {s t : Set X}, IsCompact s → IsClosed t → t ⊆ s → IsCompact tA closed subset of a compact set is a compact set.
- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 67 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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 and proof · cited by 1,282
- Set.inter_eq_self_of_subset_rightproof · cited by 39
- IsCompact.inter_rightproof · cited by 30
Cited by67
Results whose statement or proof uses this declaration.
- IsClosed.isCompactproof · cited by 43
- IsCompact.closure_of_subsetproof · cited by 15
- isCompact_sphereproof · cited by 11
- Function.locallyFinsuppWithin.finiteSupportproof · cited by 9
- Metric.isCompact_of_isClosed_isBoundedproof · cited by 8
- IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosedproof · cited by 4
- HasCompactSupport.isCompact_preimageproof · cited by 4
- exists_pos_lt_subset_ballproof · cited by 4
- rieszContentAux_image_nonemptyproof · cited by 3
- HasCompactSupport.comp_isClosedEmbeddingproof · cited by 3
- HasCompactSupport.intro'proof · cited by 3
- HasCompactSupport.iteratedFDerivproof · cited by 3