Theorems · Theorem · general topology
frontier_compl
∀ {X : Type u} [inst : TopologicalSpace X] (s : Set X), frontier sᶜ = frontier sThe complement of a set has the same frontier as the original set.
- Defined in
- Mathlib.Topology.Closure
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 65 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.
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
- Compl.complstatement and proof · cited by 2,925
- closureproof · cited by 1,254
- Set.inter_commproof · cited by 291
- compl_complproof · cited by 229
- frontierstatement · cited by 214
- frontier_eq_closure_inter_closureproof · cited by 12
Cited by10
Results whose statement or proof uses this declaration.
- frontier_lt_subset_eqproof · cited by 2
- Subalgebra.spectrum_sUnion_connectedComponentInproof · cited by 2
- Subalgebra.frontier_spectrumproof · cited by 2
- IsLowerSet.null_frontierproof · cited by 1
- ite_inter_closure_compl_eq_of_inter_frontier_eqproof · cited by 1
- IsUpperSet.null_frontierproof · cited by 1
- frontier_eq_inter_compl_interiorproof · cited by 1
- frontier_gt_subset_eqproof · cited by 0
- disjoint_frontier_iff_isOpenproof · cited by 0
- frontier_union_subsetproof · cited by 0