Theorems · Theorem · general topology

ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE

∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [T2Space Y] {x : X}
  {f g : X → Y},
  ContinuousAt f x → ContinuousAt g x → ∀ [(nhdsWithin x {x}ᶜ).NeBot], f =ᶠ[nhdsWithin x {x}ᶜ] g ↔ f =ᶠ[nhds x] g

Local identity principle for continuous maps: Two continuous maps into a Hausdorff space agree in a punctured neighborhood of a non-isolated point iff they agree in a neighborhood.

Defined in

Mathlib.Topology.Separation.Hausdorff

Cited by

6 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceT2SpaceFilter.NeBot