Theorems · Theorem · general topology
mem_closure_of_tendsto
∀ {X : Type u} [inst : TopologicalSpace X] {α : Type u_1} {x : X} {s : Set X} {f : α → X} {b : Filter α} [b.NeBot],
Filter.Tendsto f b (nhds x) → (∀ᶠ (x : α) in b, f x ∈ s) → x ∈ closure s- Defined in
- Mathlib.Topology.Neighborhoods
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceFilter.NeBot
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Filter.Eventuallystatement and proof · cited by 3,134
- closurestatement · cited by 1,254
- Filter.NeBotstatement and proof · cited by 853
- Filter.Eventually.frequentlyproof · cited by 44
- mem_closure_of_frequently_of_tendstoproof · cited by 4
Cited by19
Results whose statement or proof uses this declaration.
- stronglyMeasurable_of_tendstoproof · cited by 12
- MeasureTheory.StronglyMeasurable.isSeparable_rangeproof · cited by 10
- aestronglyMeasurable_of_tendsto_aeproof · cited by 9
- ContinuousWithinAt.mem_closure_imageproof · cited by 4
- MeasureTheory.StronglyMeasurable.measurableSet_exists_tendstoproof · cited by 3
- mem_closure_of_gauge_le_oneproof · cited by 3
- range_derivWithin_subset_closure_span_imageproof · cited by 3
- MvPowerSeries.WithPiTopology.denseRange_toMvPowerSeriesproof · cited by 2
- mem_closure_of_nonempty_tangentConeAtproof · cited by 1
- one_mem_posTangentConeAt_iff_mem_closureproof · cited by 1
- CantorScheme.ClosureAntitone.map_of_vanishingDiamproof · cited by 1
- polynomialFunctions_closure_eq_top'proof · cited by 1