Theorems · Theorem · general topology
closure_compl
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, closure sᶜ = (interior s)ᶜ- Defined in
- Mathlib.Topology.Closure
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 63 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.
Cites7
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
- Compl.complstatement and proof · cited by 2,925
- closurestatement · cited by 1,254
- interiorstatement and proof · cited by 714
- compl_complproof · cited by 229
- closure_eq_compl_interior_complproof · cited by 7
Cited by19
Results whose statement or proof uses this declaration.
- frontier_eq_closure_inter_closureproof · cited by 12
- interior_eq_empty_iff_dense_complproof · cited by 5
- IsUpperSet.interiorproof · cited by 4
- ContinuousOn.ifproof · cited by 4
- ContinuousOn.if'proof · cited by 3
- IsGδ.dense_iUnion_interior_of_closedproof · cited by 2
- Metric.PiNatEmbed.separationproof · cited by 2
- OpenPartialHomeomorph.IsImage.interiorproof · cited by 2
- exists_mem_frontier_infDist_compl_eq_distproof · cited by 2
- ContinuousMap.setOfIdeal_ofSet_eq_interiorproof · cited by 1
- frontier_eq_inter_compl_interiorproof · cited by 1
- mem_interior_iff_not_clusterPt_complproof · cited by 1