Arcwise connectedness of metric groups

DefinitionConnectedness in the sense of Hausdorff#✓fully proved

A topological space \(G\) is connected in the sense of Hausdorff when it is not the union of two disjoint nonempty closed subsets.

The empty space satisfies this condition.

def IsHausdorffConnected (G : Type*) [TopologicalSpace G] : Prop :=
  ¬ ∃ (A B : Set G), IsClosed A ∧ IsClosed B ∧ A.Nonempty ∧ B.Nonempty ∧
    Disjoint A B ∧ A ∪ B = Set.univ

Used by 1

Used byClaim of Scottish Book Problem 160, part (2)

DefinitionArcwise connected space#✓fully proved

A topological space \(G\) is arcwise connected when for any two points \(x, y \in G\) there is a continuous map \(\gamma \colon \mathbb{R} \to G\) with \(\gamma(0) = x\) and \(\gamma(1) = y\).

Restricting \(\gamma\) to \([0, 1]\) gives a path from \(x\) to \(y\), and every such path extends to \(\mathbb{R}\), so this says that any two points are joined by a path; the path need not be injective.

def IsArcwiseConnected (G : Type*) [TopologicalSpace G] : Prop :=
  ∀ x y : G, ∃ γ : ℝ → G, Continuous γ ∧ γ 0 = x ∧ γ 1 = y

Used by 1

Used byClaim of Scottish Book Problem 160, part (2)

DefinitionClaim of Scottish Book Problem 160, part (2)#✓fully proved

The claim that every metric group \(G\) (a group with a metric for which multiplication and inversion are continuous) that is connected in the sense of Hausdorff is arcwise connected.

def ScottishBookProblem160Part2Claim : Prop :=
  ∀ (G : Type*) [MetricSpace G] [Group G] [IsTopologicalGroup G],
    IsHausdorffConnected G → IsArcwiseConnected G

Uses 2 · Used by 1

Statement usesArcwise connected space · Connectedness in the sense of Hausdorff

Used byConnected metric groups need not be arcwise connected

TheoremConnected metric groups need not be arcwise connected#●ready to prove

The claim of part (2) of Scottish Book Problem 160 is false: there is a metric group \(G\) (a group with a metric for which multiplication and inversion are continuous) that is connected in the sense of Hausdorff but not arcwise connected, that is, there are points \(x, y \in G\) such that no continuous map \(\gamma \colon \mathbb{R} \to G\) has \(\gamma(0) = x\) and \(\gamma(1) = y\).

The \(p\)-adic solenoid, the inverse limit of circles under \(z \mapsto z^p\), is such a group: it is a compact connected metrizable topological group that is not path connected.

theorem_wanted scottishBook160Part2_refuted : ¬ ScottishBookProblem160Part2Claim

Sources

• Encyclopedia of Mathematics, "Solenoid" (inverse-limit construction with fᵢ (z) = z ^ nᵢ, a topological group, a one-dimensional continuum); E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I (1979), cited there; solenoids are compact metrizable spaces that are connected but not path connected.

Uses 1

Statement usesClaim of Scottish Book Problem 160, part (2)

3 fully proved1 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.