[Paper Review] A Nonseparably Connected Metric Space as a Dense Connected Graph

Michał Morayne, Michał Ryszard Wójcik|ArXiv.org|Nov 17, 2008

Data Management and Algorithms8 references3 citations

TL;DR

This paper constructs a nonseparably connected metric space as the dense, connected graph of a real-valued function satisfying Cauchy’s functional equation. By leveraging transfinite induction and Hamel basis constructions in normed spaces of size 𝔠, the authors produce a homogeneous, nowhere locally connected space where every nontrivial connected subset is nonseparable, providing a new example of a nonseparably connected metric space.

ABSTRACT

We present a connected metric space that does not contain any nontrivial separable connected subspace. Our space is a dense connected graph of a function from the real line satisfying Cauchy's equation.

Motivation & Objective

• To construct a new example of a nonseparably connected metric space—defined as a connected space with no nontrivial separable connected subsets.
• To demonstrate that such a space can arise as the graph of a function satisfying Cauchy’s equation f(x+u)=f(x)+f(u).
• To show that the space is homogeneous, not locally connected at any point, and every point is a cut point.
• To establish that the graph of the function is dense and connected in ℝ×Y for nonseparable normed spaces Y of size 𝔠.

Proposed method

• Use transfinite induction to construct a function F: ℝ → Y that intersects every closed separable subset of ℝ×Y with uncountable projection on ℝ.
• Apply Lemma 1 to ensure the function meets all required closed separable sets in the product space, based on cardinality constraints.
• Construct F as a Q-linear function over a Hamel basis, extending it to satisfy Cauchy’s equation f(x+u)=f(x)+f(u).
• Leverage Lemma 2 to prove that any such function with dense graph intersection yields a connected, dense graph in X×Y.
• Use the fact that ℝ×Y is T5 and separable subsets are hereditarily separable to ensure topological connectivity.
• Verify that the resulting space M is a topological group under addition, making it homogeneous and ensuring every point is a cut point.

Experimental results

Research questions

• RQ1 Can a nonseparably connected metric space be constructed as the graph of a function satisfying Cauchy’s equation?
• RQ2 Does such a space necessarily fail to be locally connected at any point?
• RQ3 Can the construction be carried out in normed spaces of size 𝔠, ensuring the function is additive and the graph is dense and connected?
• RQ4 Is there a complete nonseparably connected metric space, or does such a space require the Axiom of Choice?
• RQ5 Can the graph of a Cauchy-functional function be both dense and connected in ℝ×Y for nonseparable Y?

Key findings

• The graph of a function F: ℝ → l∞ satisfying Cauchy’s equation f(x+u)=f(x)+f(u) is dense and connected in ℝ×l∞.
• The resulting metric space M is nonseparably connected: every nontrivial connected subset is nonseparable.
• Every point in M is a cut point, meaning M∖{p} is disconnected for every p∈M.
• The space M is not locally connected at any point, due to the function being discontinuous everywhere.
• M is a topological group under the operation (x,F(x))+(u,F(u))=(x+u,F(x+u)), making it homogeneous.
• The construction relies on the Axiom of Choice and cannot be carried out in ZF alone, raising the open question of whether such a space exists in ZF.