Theorems · Theorem · general topology

ContinuousOn.preimage_connectedComponentIn

∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β} {F : Set β},
  ContinuousOn f (f ⁻¹' F) →
    ∀ (y : β), f ⁻¹' connectedComponentIn F y = ⋃ x ∈ f ⁻¹' connectedComponentIn F y, connectedComponentIn (f ⁻¹' F) x

The preimage of a connected component of F is the union of the connected components of f ⁻¹' F at the points of that preimage.

Defined in

Mathlib.Topology.Connected.Basic

Cited by

2 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace