Theorems · Theorem · general topology
IsClosed.closure_subset
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsClosed s → closure s ⊆ s- Defined in
- Mathlib.Topology.Closure
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 62 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
- closurestatement · cited by 1,254
- closure_minimalproof · cited by 94
- Set.Subset.reflproof · cited by 66
Cited by11
Results whose statement or proof uses this declaration.
- closure_subset_iff_isClosedproof · cited by 11
- ContinuousWithinAt.closure_leproof · cited by 5
- Filter.Frequently.mem_of_closedproof · cited by 5
- IsGLB.mem_of_isClosedproof · cited by 4
- IsLUB.mem_of_isClosedproof · cited by 3
- isClosed_iff_derivedSet_subsetproof · cited by 3
- isMIntegralCurveOn_Ioo_eqOn_of_contMDiffproof · cited by 2
- LocallyFinite.continuousOn_iUnionproof · cited by 1
- Metric.isBounded_closure_of_isBoundedproof · cited by 1
- Metric.closure_thickening_subset_cthickeningproof · cited by 0
- continuous_iff_image_closure_subset_closure_imageproof · cited by 0