Kourovka Notebook Problem 21.103

Definitions and mathematical statement

A Hausdorff group topology $\tau$ on an abstract group $G$ makes multiplication and inversion continuous and separates distinct points. The topological group $(G,\tau)$ is minimal if every Hausdorff group topology $\sigma\subseteq\tau$ is equal to $\tau$; thus it admits no strictly coarser Hausdorff group topology.

For an identity neighborhood $U$, a basic entourage of the two-sided uniformity is \[ E_U=\{(x,y)\in G^2:x^{-1}y\in U\text{ and }xy^{-1}\in U\}. \] A net $(x_\alpha)$ is two-sided Cauchy if for every $U$, all sufficiently late pairs $(x_\alpha,x_\beta)$ lie in $E_U$. The group is Raikov complete when every such Cauchy net converges in $G$.

Let $I$ be an arbitrary index set and let $(G_i,\tau_i)$ be a Raikov complete minimal Hausdorff topological group for every $i\in I$. Equip \[ G=\prod_{i\in I}G_i \] with coordinatewise multiplication and the product topology $\tau$. A basic open neighborhood restricts only finitely many coordinates. Must $(G,\tau)$ be minimal?

Explicitly, the target is that every Hausdorff group topology on this underlying product group contained in $\tau$ must coincide with $\tau$. The index set may be infinite or uncountable, and the factors need not be compact, metrizable, or center-free. Completeness refers to the two-sided uniformity specified above, so the hypothesis is a property of each given factor topology.

Short English statement

Is every arbitrary product of Raikov complete minimal Hausdorff topological groups again minimal?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2612 (KOU-21.103), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.103. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.