Open Problems in Topological Algebra — Problem 7 (Protasov)

Definitions and mathematical statement

A space is zero-dimensional if it has a base consisting of sets that are both open and closed. It is first-countable if each point has a countable neighborhood base, and homogeneous if for every pair of points $x,y$ there is a homeomorphism of the space onto itself carrying $x$ to $y$.

A left topological group structure on a nonempty topological space $X$ is a group operation $\cdot\colon X\times X\to X$ such that, with the given topology on $X$, every left translation \[ L_a\colon X\longrightarrow X,\qquad L_a(x)=a\cdot x, \] is continuous for every $a\in X$. Each such translation is then a homeomorphism, since its inverse is $L_{a^{-1}}$. Joint continuity of multiplication and continuity of inversion are not required by this definition.

General question, reported negatively answered in the supplied source. Does every nonempty zero-dimensional compact homogeneous space admit a left topological group structure? The supplied record reports that Banakh answered this question negatively in the 2008 paper cited below; its title specifies a homogeneous first-countable zero-dimensional compactum that fails to be a left topological group. This historical resolution is reported from the supplied record, not independently verified here.

Further question retained from the source. For every nonempty first-countable zero-dimensional compact space $K$, does its countable power \[ K^{\omega}=\prod_{n=0}^{\infty}K, \] endowed with the product topology, admit a left topological group structure? No homogeneity assumption on $K$ itself is stated in this question.

Unresolved qualifications. The supplied excerpt does not specify whether Hausdorffness is implicit in the word "compact"; this convention must be confirmed before fixing the precise class of spaces quantified over. The nonempty-space formulation above makes explicit the substantive reading of the questions: the excerpt does not expressly exclude the empty space, which cannot carry a group structure. The registry labels the record open despite including a reported negative answer to the general question; it does not explicitly clarify the scope of that status label.

Short English statement

The source asks whether zero-dimensional compact homogeneous spaces can carry group operations for which all left translations are continuous. It reports a negative answer to that general question, then asks whether the countable product of every first-countable zero-dimensional compact space can carry such an operation. The supplied open label is not an independently verified status assessment; separation and nonemptiness conventions require review.

Sources

• Local UnsolvedMath registry, permanent ID $800733$, source problem code catalogue entry, title "Open Problems in Topological Algebra --- Problem 7 (Protasov)." The registry lists Banakh and Choban as proposers, gives the year 2012, and labels the record open.
• Banakh and Choban, Some Open Problems in Topological Algebra (2012), Problem 7. Supplied source URL: https://arxiv.org/abs/1202.1619. The supplied background attributes the problem or its context to Protasov. It records extraction from arXiv LaTeX and an access date of 2026-10-16; these are supplied provenance details, not a claim of independent access or verification.
• T. Banakh, A homogeneous first-countable zero-dimensional compactum failing to be a left-topological group, Mat. Stud. 29 (2008), 215--217. The supplied statement cites this paper as giving a negative answer to the general question. No separate URL for this paper is supplied.