Theorems · Theorem · general topology
IsOpen.mem_nhds_iff
∀ {X : Type u} [inst : TopologicalSpace X] {x : X} {s : Set X}, IsOpen s → (s ∈ nhds x ↔ x ∈ s)- Defined in
- Mathlib.Topology.Neighborhoods
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 60 from the axioms · uses propext, Quot.sound
- Assumes
- TopologicalSpace
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
- TopologicalSpacestatement and proof · cited by 24,529
- Filterstatement · cited by 8,121
- nhdsstatement · cited by 5,554
- IsOpenstatement and proof · cited by 2,400
- Set.Subset.rflproof · cited by 255
- mem_of_mem_nhdsproof · cited by 126
- mem_nhds_iffproof · cited by 67
Cited by10
Results whose statement or proof uses this declaration.
- ENNReal.continuousAt_toRealproof · cited by 6
- compl_singleton_mem_nhds_iffproof · cited by 4
- UniformSpace.hausdorff.isOpen_inter_nonempty_of_isOpenproof · cited by 4
- lebesgue_number_lemmaproof · cited by 4
- nhds_basis_opens'proof · cited by 3
- Complex.norm_eqOn_of_isPreconnected_of_isMaxOnproof · cited by 3
- UniformSpace.nhds_basis_clopensproof · cited by 1
- completelyRegularSpace_iInfproof · cited by 1
- Distribution.dsupport_deltaproof · cited by 0
- Distribution.TemperedDistribution.dsupport_deltaproof · cited by 0