Theorems · Theorem · general topology

Topology.IsInducing.isPathConnected_iff

∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {F : Set X} {f : X → Y},
  Topology.IsInducing f → (IsPathConnected F ↔ IsPathConnected (f '' F))

If f : X → Y is an inducing map, f(F) is path-connected iff F is.

Defined in

Mathlib.Topology.Connected.PathConnected

Cited by

4 results in Mathlib

Foundations

Depth 128 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace