Theorems · Theorem · general topology
IsClosed.mem_of_frequently_of_tendsto
∀ {X : Type u} [inst : TopologicalSpace X] {α : Type u_1} {x : X} {s : Set X} {f : α → X} {b : Filter α},
IsClosed s → (∃ᶠ (x : α) in b, f x ∈ s) → Filter.Tendsto f b (nhds x) → x ∈ s- Defined in
- Mathlib.Topology.Neighborhoods
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 66 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.
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
- TopologicalSpacestatement and proof · cited by 24,529
- Filterstatement and proof · cited by 8,121
- nhdsstatement and proof · cited by 5,554
- Filter.Tendstostatement and proof · cited by 3,814
- IsClosedstatement and proof · cited by 1,639
- Filter.Frequentlystatement and proof · cited by 414
- Filter.Tendsto.frequentlyproof · cited by 12
- Filter.Frequently.mem_of_closedproof · cited by 5
Cited by5
Results whose statement or proof uses this declaration.
- IsClosed.mem_of_tendstoproof · cited by 32
- le_of_tendsto_of_tendsto_of_frequentlyproof · cited by 2
- isClosed_iUnion_closure_singleton_of_not_tendstoproof · cited by 1
- ge_of_tendsto_of_frequentlyproof · cited by 0
- le_of_tendsto_of_frequentlyproof · cited by 0