Theorems · Theorem · general topology
IsOpen.eventually_mem
∀ {X : Type u} [inst : TopologicalSpace X] {x : X} {s : Set X}, IsOpen s → x ∈ s → ∀ᶠ (x : X) in nhds x, x ∈ s- Defined in
- Mathlib.Topology.Neighborhoods
- Cited by
- 24 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.
Cites6
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
- nhdsstatement · cited by 5,554
- Filter.Eventuallystatement · cited by 3,134
- IsOpenstatement and proof · cited by 2,400
- IsOpen.mem_nhdsproof · cited by 470
Cited by24
Results whose statement or proof uses this declaration.
- Filter.Tendsto.eventually_neproof · cited by 11
- OpenPartialHomeomorph.eventually_right_inverseproof · cited by 9
- eventually_ne_nhdsproof · cited by 7
- OpenPartialHomeomorph.eventually_left_inverseproof · cited by 7
- analyticAt_clogproof · cited by 6
- Complex.hasStrictDerivAt_const_cpowproof · cited by 6
- ConvexOn.continuousOn_tfaeproof · cited by 3
- ProbabilityTheory.hasDerivAt_integral_pow_mul_exp_realproof · cited by 3
- Complex.hasStrictFDerivAt_cpowproof · cited by 3
- lowerSemicontinuousOn_iff_isClosed_epigraphproof · cited by 2
- Continuous.exists_contMDiff_approx_and_eqOnproof · cited by 2
- AnalyticAt.eventually_constant_or_nhds_le_map_nhds_auxproof · cited by 1