Theorems · Theorem · general topology
nhds_inter_eq_singleton_of_mem_discrete
∀ {X : Type u_1} [inst : TopologicalSpace X] {s : Set X}, IsDiscrete s → ∀ {x : X}, x ∈ s → ∃ U ∈ nhds x, U ∩ s = {x}A point x in a discrete subset s of a topological space admits a neighbourhood
that only meets s at x.
- Defined in
- Mathlib.Topology.Separation.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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 and proof · cited by 5,554
- Filter.basis_setsproof · cited by 105
- IsDiscretestatement and proof · cited by 86
- Filter.HasBasis.exists_inter_eq_singleton_of_mem_discreteproof · cited by 3
Cited by4
Results whose statement or proof uses this declaration.
- IsDiscrete.exists_nhds_eq_zero_of_image_addLeft_inter_nonemptyproof · cited by 2
- IsDiscrete.exists_nhds_eq_one_of_image_mulLeft_inter_nonemptyproof · cited by 2
- isOpen_inter_eq_singleton_of_mem_discreteproof · cited by 1
- disjoint_nhdsWithin_of_mem_discreteproof · cited by 0