Theorems · Theorem · general topology

IsPreconnected.constant_of_mapsTo

∀ {α : Type u} [inst : TopologicalSpace α] {S : Set α},
  IsPreconnected S →
    ∀ {β : Type u_3} [inst_1 : TopologicalSpace β] {T : Set β},
      IsDiscrete T → ∀ {f : α → β}, ContinuousOn f S → Set.MapsTo f S T → ∀ {x y : α}, x ∈ S → y ∈ S → f x = f y

Refinement of IsPreconnected.constant only assuming the map factors through a discrete subset of the target.

Defined in

Mathlib.Topology.Connected.TotallyDisconnected

Cited by

1 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace