Theorems · Theorem · general topology

OpenPartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_left

∀ {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
  [inst_2 : TopologicalSpace Z] (e : OpenPartialHomeomorph X Y) {f : Z → X} {s : Set Z} {x : Z},
  f x ∈ e.source → f ⁻¹' e.source ∈ nhdsWithin x s → (ContinuousWithinAt f s x ↔ ContinuousWithinAt (↑e ∘ f) s x)

Continuity within a set at a point can be read under left composition with a local homeomorphism if a neighborhood of the initial point is sent to the source of the local homeomorphism

Defined in

Mathlib.Topology.OpenPartialHomeomorph.Continuity

Cited by

2 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceTopologicalSpace