Theorems · Theorem · general topology
isClopen_iff_frontier_eq_empty
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsClopen s ↔ frontier s = ∅- Defined in
- Mathlib.Topology.Clopen
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 67 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.
Cites15
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
- LE.le.transproof · cited by 3,151
- IsOpenproof · cited by 2,400
- closureproof · cited by 1,254
- interiorproof · cited by 714
- LE.le.antisymmproof · cited by 507
- subset_closureproof · cited by 309
- frontierstatement and proof · cited by 214
- IsClopenstatement · cited by 189
- interior_subsetproof · cited by 171
- Eq.subsetproof · cited by 124
Cited by6
Results whose statement or proof uses this declaration.
- IsClopen.frontier_eqproof · cited by 2
- IsClopen.continuous_indicatorproof · cited by 2
- IsUltrametricDist.frontier_ball_eq_emptyproof · cited by 0
- IsUltrametricDist.frontier_closedBall_eq_emptyproof · cited by 0
- frontier_eq_empty_iffproof · cited by 0
- IsClopen.continuous_mulIndicatorproof · cited by 0