Mathoverflow 235893

Reference: mathoverflow/235893 asked by user Willie Wong

def Mathoverflow235893.IsConnectedMap {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) : Prop

For topological spaces $X$ and $Y$ we say a function $f : X → Y$ is connected is it sends connected sets to connected sets.

Equations

• Mathoverflow235893.IsConnectedMap f = ∀ ⦃s : Set X⦄, IsConnected s → IsConnected (f '' s)

theorem Mathoverflow235893.Continuous.isConnectedMap {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) : IsConnectedMap f

By a standard result, every continuous map is connected

theorem Mathoverflow235893.mathoverflow_235893 : sorry ↔ ∀ n > 0, ∃ (f : EuclideanSpace ℝ (Fin n) ≃ EuclideanSpace ℝ (Fin n)), IsConnectedMap ⇑f ∧ ¬IsConnectedMap ⇑f.symm

Does there exist a bijection $f : ℝ^n → ℝ^n$ such that $f$ is connected but the inverse is not?

theorem Mathoverflow235893.mathoverflow_260589 : ∃ (f : ℝ ≃ EuclideanSpace ℝ (Fin 2)), IsConnectedMap ⇑f ∧ ¬IsConnectedMap ⇑f.symm

There exists a connected bijection ℝ → ℝ^2 where the inverse is not connected, proven in mathoverflow/260589 by user Gro-Tsen.