Theorems · Theorem · general topology
Metric.ball_subset_closedBall
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, Metric.ball x ε ⊆ Metric.closedBall x ε- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 46 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.
Cites7
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
- le_of_ltproof · cited by 1,175
- Metric.ballstatement and proof · cited by 735
- Metric.closedBallstatement · cited by 704
- Metric.mem_closedBallproof · cited by 42
Cited by46
Results whose statement or proof uses this declaration.
- Metric.closedBall_mem_nhdsproof · cited by 20
- Metric.isBounded_ballproof · cited by 15
- Metric.closure_ball_subset_closedBallproof · cited by 10
- Metric.measure_closedBall_posproof · cited by 8
- Topology.RelCWComplex.openCell_subset_closedCellproof · cited by 8
- div_le_egauge_ballproof · cited by 4
- Metric.closedBall_mem_nhds_of_memproof · cited by 4
- exists_pos_lt_subset_ballproof · cited by 4
- Metric.ball_subset_interior_closedBallproof · cited by 4
- Bornology.IsBounded.subset_closedBall_ltproof · cited by 3
- SmoothBumpFunction.tsupport_subset_symm_image_closedBallproof · cited by 3
- MeromorphicOn.circleAverage_log_normproof · cited by 3