Continuum theory and infinite-dimensional topology

Compact connected spaces, Peano continua and space-filling curves, absolute retracts, hyperspaces, the Hilbert cube and fixed-point properties.

TheoremHahn–Mazurkiewicz theorem#✓fully proved

Let \(X\) be a nonempty topological space that is compact, connected, locally connected, Hausdorff and second countable. Then there is a continuous surjection from the closed unit interval \([0,1]\) onto \(X\).

public theorem hahnMazurkiewicz
    {X : Type*} [TopologicalSpace X] [CompactSpace X] [ConnectedSpace X]
    [LocallyConnectedSpace X] [T2Space X] [SecondCountableTopology X] [Nonempty X] :
    ∃ (f : unitInterval → X), Continuous f ∧ Function.Surjective f

Sources

H. Hahn, Jahresber. DMV 23 (1914); S. Mazurkiewicz, Fund. Math. 1 (1920) (announcement 1913); Willard, General Topology; Munkres, Topology 2nd ed., second-countable Hausdorff compact formulation.

TheoremPeano's space-filling curve#✓fully proved

There exists a continuous surjection from the closed unit interval to its square.

theorem peano_curve_exists :
    ∃ (f : C(unitInterval, unitInterval × unitInterval)), Function.Surjective f

Sources

G. Peano, "Sur une courbe, qui remplit toute une aire plane", Mathematische Annalen 36 (1890), 157-160.

TheoremUnit spheres of infinite-dimensional Banach spaces are unicoherent#●ready to prove

Let \(E\) be a real Banach space that is not finite-dimensional, and let \(S = \{x \in E : \|x\| = 1\}\) be its unit sphere. If \(A\) and \(B\) are closed connected subsets of \(S\) with \(A \cup B = S\), then \(A \cap B\) is connected. Here connected sets are nonempty by definition.

This answers Banach's Problem 46 of the Scottish Book: is the sphere in a space of type (B) uni-coherent, that is, whenever it is the union of two continua \(A\) and \(B\), is \(A \cap B\) connected? Continua are read as closed connected sets, the meaning of uni-coherence for non-compact spaces: in infinite dimensions the sphere is not compact, so two compact sets could not cover it. The statement is restricted to infinite-dimensional spaces; in dimension \(1\) the sphere is two points and in dimension \(2\) it is a circle, and neither is uni-coherent (two semicircles cover the circle and meet in two points). In infinite dimensions the sphere is an absolute retract, and the affirmative answer follows from the theorems of Borsuk and Eilenberg recorded in the addendum.

theorem_wanted sphere_in_banach_space_is_unicoherent :
    ∀ (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E],
      ¬ Module.Finite ℝ E →
      ∀ A B : Set E, A ⊆ Metric.sphere (0 : E) 1 → B ⊆ Metric.sphere (0 : E) 1 →
        A ∪ B = Metric.sphere (0 : E) 1 → IsClosed A → IsConnected A →
        IsClosed B → IsConnected B → IsConnected (A ∩ B)

Sources

S. Banach, Problem 46 of the Scottish Book; addendum in the Scottish Book citing Borsuk and Eilenberg, Fund. Math. 26, p. 61; see the commentary in R. D. Mauldin (ed.), The Scottish Book, 2nd ed. (Birkhäuser, 2015).

2 fully proved1 ready to prove

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