Theorems · Theorem · general topology
Metric.closedBall_mem_nhds
∀ {α : Type u} [inst : PseudoMetricSpace α] (x : α) {ε : ℝ}, 0 < ε → Metric.closedBall x ε ∈ nhds x- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 20 results in Mathlib
- Foundations
- Depth 120 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.
Cites9
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
- Filterstatement · cited by 8,121
- nhdsstatement · cited by 5,554
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.closedBallstatement · cited by 704
- Filter.mem_of_supersetproof · cited by 308
- Metric.ball_mem_nhdsproof · cited by 64
- Metric.ball_subset_closedBallproof · cited by 46
Cited by20
Results whose statement or proof uses this declaration.
- upperHemicontinuous_spectrumproof · cited by 4
- Filter.Tendsto.integral_sub_linear_isLittleO_aeproof · cited by 3
- FiniteDimensional.of_isCompact_closedBall₀proof · cited by 2
- MeasureTheory.IntegrableOn.hasBoxIntegralproof · cited by 2
- HasFDerivWithinAt.curveIntegral_segment_source'proof · cited by 2
- SmoothBumpFunction.nhdsWithin_range_basisproof · cited by 1
- eventually_singleton_add_smul_subsetproof · cited by 1
- continuous_cfcHomSuperset_leftproof · cited by 1
- continuous_cfcₙHomSuperset_leftproof · cited by 1
- isCompactOperator_iff_isCompact_closure_image_closedBallproof · cited by 1
- tendsto_integral_comp_smul_smul_of_integrableproof · cited by 1