Theorems · Theorem · general topology
interior_frontier
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsClosed s → interior (frontier s) = ∅The frontier of a closed set has no interior point.
- Defined in
- Mathlib.Topology.Closure
- Cited by
- 3 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.
Cites12
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
- interiorstatement and proof · cited by 714
- frontierstatement and proof · cited by 214
- interior_subsetproof · cited by 171
- Set.sdiff_subsetproof · cited by 156
- Set.subset_interproof · cited by 74
- Set.subset_empty_iffproof · cited by 40
- interior_monoproof · cited by 38
- Set.inter_sdiff_selfproof · cited by 4
- IsClosed.frontier_eqproof · cited by 4
Cited by3
Results whose statement or proof uses this declaration.
- IsGδ.dense_iUnion_interior_of_closedproof · cited by 2
- interior_sphereproof · cited by 1
- interior_sphere'proof · cited by 1