Theorems · Theorem · general topology
frontier_subset_iff_isClosed
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, frontier s ⊆ s ↔ IsClosed s- Defined in
- Mathlib.Topology.Closure
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 68 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.
Cites9
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
- closureproof · cited by 1,254
- frontierstatement · cited by 214
- interior_subsetproof · cited by 171
- Set.sdiff_subset_iffproof · cited by 15
- Set.union_eq_rightproof · cited by 14
- closure_subset_iff_isClosedproof · cited by 11
Cited by2
Results whose statement or proof uses this declaration.
- IsClosed.frontier_subsetproof · cited by 2
- disjoint_frontier_iff_isOpenproof · cited by 0