Theorems · Theorem · general topology
IsClosed.closure_eq
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsClosed s → closure s = s- Defined in
- Mathlib.Topology.Closure
- Cited by
- 139 results in Mathlib
- Foundations
- Depth 62 from the axioms, rests on 681 definitions · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
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
- closurestatement · cited by 1,254
- subset_closureproof · cited by 309
- Set.Subset.antisymmproof · cited by 213
- closure_minimalproof · cited by 94
- Set.Subset.reflproof · cited by 66
Cited by139
Results whose statement or proof uses this declaration.
- Measurable.stronglyMeasurableproof · cited by 47
- Function.Surjective.denseRangeproof · cited by 21
- uniqueDiffWithinAt_univproof · cited by 16
- isClosed_propertyproof · cited by 13
- Continuous.closure_preimage_subsetproof · cited by 12
- closure_eq_iff_isClosedproof · cited by 12
- closure_emptyproof · cited by 11
- IsClosed.prodproof · cited by 10
- closure_closureproof · cited by 9
- closure_singletonproof · cited by 8
- genericPoint_specproof · cited by 7
- regularSpace_TFAEproof · cited by 6