Theorems · Theorem · general topology
Metric.isOpen_ball
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, IsOpen (Metric.ball x ε)- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 63 results in Mathlib
- Foundations
- Depth 118 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.
- Realstatement and proof · cited by 25,697
- IsOpenstatement · cited by 2,400
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.ballstatement · cited by 735
- Metric.isOpen_iffproof · cited by 22
- Metric.exists_ball_subset_ballproof · cited by 2
Cited by63
Results whose statement or proof uses this declaration.
- Metric.ball_mem_nhdsproof · cited by 64
- measurableSet_ballproof · cited by 15
- ExistsContDiffBumpBase.u_existsproof · cited by 5
- Metric.measure_ball_posproof · cited by 5
- InnerProductSpace.HarmonicOnNhd.exists_analyticOnNhd_ball_re_eqproof · cited by 4
- Metric.closedBall_mem_nhds_of_memproof · cited by 4
- Metric.ball_subset_interior_closedBallproof · cited by 4
- Complex.one_add_cpow_hasFPowerSeriesOnBall_zeroproof · cited by 3
- IsOpen.isOpen_inter_preimage_of_fderiv_eq_zeroproof · cited by 3
- DiffContOnCl.circleIntegral_one_div_sub_center_pow_smulproof · cited by 3
- eventually_residual_liouvilleproof · cited by 2