Theorems · Theorem · general topology
IsUltrametricDist.isClopen_closedBall
∀ {X : Type u_1} [inst : PseudoMetricSpace X] [IsUltrametricDist X] (x : X) {r : ℝ},
r ≠ 0 → IsClopen (Metric.closedBall x r)- Defined in
- Mathlib.Topology.MetricSpace.Ultra.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.closedBallstatement · cited by 704
- IsClopenstatement · cited by 189
- IsUltrametricDiststatement and proof · cited by 177
- Metric.isClosed_closedBallproof · cited by 28
- IsUltrametricDist.isOpen_closedBallproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- IsUltrametricDist.frontier_closedBall_eq_emptyproof · cited by 0
- Padic.withValUniformEquiv_norm_le_one_iffproof · cited by 0