Theorems · Theorem · general topology

Continuous.preimage_connectedComponent

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

The preimage of a connected component is the union of the connected components at the points of that preimage.

Defined in

Mathlib.Topology.Connected.Basic

Cited by

0 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace