Theorems · Theorem · general topology
Metric.sphere_subset_closedBall
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, Metric.sphere x ε ⊆ Metric.closedBall x ε- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.closedBallstatement · cited by 704
- Metric.spherestatement · cited by 371
- le_of_eqproof · cited by 366
Cited by24
Results whose statement or proof uses this declaration.
- isCompact_sphereproof · cited by 11
- NormedSpace.sphere_nonemptyproof · cited by 8
- InnerProductSpace.HarmonicOnNhd.circleAverage_eqproof · cited by 5
- Topology.RelCWComplex.cellFrontier_subset_closedCellproof · cited by 4
- MeromorphicOn.circleAverage_log_normproof · cited by 3
- Complex.hasFPowerSeriesOnBall_of_differentiable_off_countableproof · cited by 3
- Metric.diam_sphere_eqproof · cited by 2
- Topology.RelCWComplex.isCompact_cellFrontierproof · cited by 2
- AnalyticAt.eventually_constant_or_nhds_le_map_nhds_auxproof · cited by 1