Theorems · Theorem · general topology
UniformSpace.mem_nhds_iff
∀ {α : Type ua} [inst : UniformSpace α] {x : α} {s : Set α}, s ∈ nhds x ↔ ∃ V ∈ uniformity α, UniformSpace.ball x V ⊆ s- Defined in
- Mathlib.Topology.UniformSpace.Defs
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- UniformSpace
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 and proof · cited by 53,352
- Filterstatement and proof · cited by 8,121
- nhdsstatement · cited by 5,554
- Set.preimageproof · cited by 4,946
- UniformSpacestatement and proof · cited by 2,040
- uniformitystatement and proof · cited by 765
- UniformSpace.ballstatement and proof · cited by 113
- nhds_eq_comap_uniformityproof · cited by 19
- Filter.mem_comapproof · cited by 13
Cited by7
Results whose statement or proof uses this declaration.
- UniformSpace.ball_mem_nhdsproof · cited by 16
- UniformSpace.hausdorff.isOpen_inter_nonempty_of_isOpenproof · cited by 4
- Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAtproof · cited by 2
- TendstoUniformlyOn.lowerHemicontinuousOnproof · cited by 1
- IsUltraUniformity.mem_nhds_iff_symm_transproof · cited by 0
- UniformSpace.mem_nhds_iff_symmproof · cited by 0
- UniformSpace.completelyNormalSpace_of_hasAntitoneBasisproof · cited by 0