Theorems · Theorem · general topology
IsUltrametricDist.isOpen_closedBall
∀ {X : Type u_1} [inst : PseudoMetricSpace X] [IsUltrametricDist X] (x : X) {r : ℝ},
r ≠ 0 → IsOpen (Metric.closedBall x r)- Defined in
- Mathlib.Topology.MetricSpace.Ultra.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- IsOpenstatement and proof · cited by 2,400
- Disjointproof · cited by 2,201
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.ballproof · cited by 735
- Metric.closedBallstatement and proof · cited by 704
- IsUltrametricDiststatement and proof · cited by 177
- LT.lt.not_gtproof · cited by 54
- lt_or_gt_of_neproof · cited by 41
- Metric.isOpen_iffproof · cited by 22
- Metric.closedBall_eq_emptyproof · cited by 15
- IsUltrametricDist.closedBall_eq_of_memproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- IsUltrametricDist.isClopen_closedBallproof · cited by 2
- IsUltrametricDist.isOpen_sphereproof · cited by 1
- PadicInt.isOpenEmbedding_coeproof · cited by 0