Theorems · Theorem · general topology
IsUltrametricDist.isOpen_sphere
∀ {X : Type u_1} [inst : PseudoMetricSpace X] [IsUltrametricDist X] (x : X) {r : ℝ}, r ≠ 0 → IsOpen (Metric.sphere x r)- Defined in
- Mathlib.Topology.MetricSpace.Ultra.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Realstatement and proof · cited by 25,697
- IsOpenstatement and proof · cited by 2,400
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.spherestatement · cited by 371
- IsUltrametricDiststatement and proof · cited by 177
- IsClosed.isOpen_complproof · cited by 126
- IsOpen.interproof · cited by 98
- sdiff_eqproof · cited by 18
- Metric.closedBall_sdiff_ballproof · cited by 10
- IsUltrametricDist.isOpen_closedBallproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- IsUltrametricDist.isClopen_sphereproof · cited by 0