Theorems · Theorem · general topology

tendstoLocallyUniformlyOn_of_forall_exists_nhds

∀ {α : Type u_1} {β : Type u_2} {ι : Type u_4} [inst : TopologicalSpace α] [inst_1 : UniformSpace β] {F : ι → α → β}
  {f : α → β} {s : Set α} {p : Filter ι},
  (∀ x ∈ s, ∃ t ∈ nhdsWithin x s, TendstoUniformlyOn F f p t) → TendstoLocallyUniformlyOn F f p s

If every x ∈ s has a neighbourhood within s on which F i tends uniformly to f, then F i tends locally uniformly on s to f. Note this is not a tautology, since our definition of TendstoLocallyUniformlyOn is slightly more general (although the conditions are equivalent if β is locally compact and s is open, see tendstoLocallyUniformlyOn_TFAE).

Defined in

Mathlib.Topology.UniformSpace.LocallyUniformConvergence

Cited by

3 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceUniformSpace