Theorems · Theorem · general topology
compl_singleton_mem_nhds_iff
∀ {X : Type u_1} [inst : TopologicalSpace X] [T1Space X] {x y : X}, {x}ᶜ ∈ nhds y ↔ y ≠ x- Defined in
- Mathlib.Topology.Separation.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 61 from the axioms · uses propext, Quot.sound
- Assumes
- TopologicalSpaceT1Space
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 · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Filterstatement · cited by 8,121
- nhdsstatement · cited by 5,554
- Compl.complstatement · cited by 2,925
- T1Spacestatement and proof · cited by 275
- isOpen_compl_singletonproof · cited by 27
- IsOpen.mem_nhds_iffproof · cited by 10
Cited by4
Results whose statement or proof uses this declaration.
- compl_singleton_mem_nhdsproof · cited by 10
- EReal.continuous_toENNRealproof · cited by 4
- meromorphicNFAt_smul_iff_right_of_analyticAtproof · cited by 3
- analyticAt_iff_analytic_fun_smulproof · cited by 1