Theorems · Theorem · general topology

ContinuousAt.ne_iff_eventually_ne

∀ {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 → (f x ≠ g x ↔ ∀ᶠ (x : X) in nhds x, f x ≠ g x)

Two continuous maps into a Hausdorff space disagree at a point iff they disagree in a neighborhood.

Defined in

Mathlib.Topology.Separation.Hausdorff

Cited by

2 results in Mathlib

Foundations

Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceT2Space