Theorems · Theorem · general topology
Metric.mem_uniformity_dist
∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set (α × α)},
s ∈ uniformity α ↔ ∃ ε > 0, ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ s- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 114 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.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement · cited by 25,697
- Filterstatement · cited by 8,121
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement · cited by 1,539
- uniformitystatement · cited by 765
- Metric.uniformity_basis_distproof · cited by 25
- Filter.HasBasis.mem_uniformity_iffproof · cited by 4
Cited by8
Results whose statement or proof uses this declaration.
- Metric.dist_mem_uniformityproof · cited by 23
- Metric.tendstoUniformlyOn_iffproof · cited by 14
- Metric.tendstoUniformlyOnFilter_iffproof · cited by 3
- CauSeq.cauchySeqproof · cited by 1
- Metric.uniformCauchySeqOn_iffproof · cited by 0
- UniformSpace.Completion.mem_uniformity_distproof · cited by 0
- Metric.tendstoLocallyUniformlyOn_iffproof · cited by 0
- Metric.equicontinuousAt_iff_pairproof · cited by 0