A Nonseparably Connected Metric Space as a Dense Connected Graph

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.

💡 Research Summary

The paper addresses the longstanding problem of exhibiting a connected metric space that contains no non‑trivial separable connected subspaces—a property known as non‑separably connectedness. While previous constructions relied on intricate fractal sets, high‑dimensional manifolds, or additional set‑theoretic assumptions (e.g., Continuum Hypothesis), the authors present a remarkably simple example built from the graph of a single real‑valued function on the real line.

The central object is a function (f:\mathbb{R}\to\mathbb{R}) satisfying Cauchy’s functional equation (f(x+y)=f(x)+f(y)). By invoking the axiom of choice, one can select a Hamel basis of (\mathbb{R}) over (\mathbb{Q}) and define (f) to be (\mathbb{Q})-linear but not continuous. Such a function is highly pathological: it is not measurable, it fails to be bounded on any interval, and, crucially for the construction, its graph (G_f={(x,f(x))\mid x\in\mathbb{R}}) is dense in the Euclidean plane (\mathbb{R}^2).

The authors equip (G_f) with the metric \