Theorems · Theorem · general topology
frontier_eq_closure_inter_closure
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, frontier s = closure s ∩ closure sᶜ- Defined in
- Mathlib.Topology.Closure
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 64 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
- closurestatement and proof · cited by 1,254
- interiorproof · cited by 714
- frontierstatement and proof · cited by 214
- Set.sdiff_eqproof · cited by 59
- closure_complproof · cited by 19
Cited by12
Results whose statement or proof uses this declaration.
- frontier_complproof · cited by 10
- frontier_le_subset_eqproof · cited by 6
- IsOpenMap.preimage_frontier_eq_frontier_preimageproof · cited by 6
- frontier_inter_subsetproof · cited by 4
- isClosed_frontierproof · cited by 2
- frontier_ge_subset_eqproof · cited by 2
- Subalgebra.frontier_subset_frontierproof · cited by 1
- SuccOrder.isSuccLimit_of_mem_frontierproof · cited by 1
- SeparatedNhds.of_isClosed_isCompact_closure_compl_isClosedproof · cited by 1
- PredOrder.isPredLimit_of_mem_frontierproof · cited by 0
- IsCompact.exists_mem_frontier_infDist_compl_eq_distproof · cited by 0
- IsOpenMap.preimage_frontier_subset_frontier_preimageproof · cited by 0