Open Problems in Topological Algebra — Problem 5 (Protasov)

Definitions and mathematical statement

Let $X$ be a topological space. The supplied excerpt uses $S(X)$ without defining its elements or any topology on it. For a subgroup $H\leq S(X)$, write $h(x)$ for its indicated action on $X$. The action is transitive if \[ \forall x,y\in X\quad \exists h\in H\quad h(x)=y. \] Following the supplied definition, $H$ is distal if \[ \forall x_1,x_2\in X\text{ with }x_1\ne x_2,\quad \forall x\in X,\quad \exists U\text{ a neighborhood of }x,\quad \forall h\in H,\quad \{h(x_1),h(x_2)\}\not\subseteq U. \] Here a neighborhood of $x$ means a set containing an open set that contains $x$. Thus the chosen neighborhood may depend on $x_1,x_2,x$, but must work for every $h\in H$.

The question is: for every compact space $X$ and every distal subgroup $H\leq S(X)$ acting transitively on $X$, must $H$ be left bounded?

The terms $S(X)$ and left bounded must retain their intended source meanings, which are not specified in the supplied excerpt. In particular, no topology or other structure defining left boundedness is supplied, so the conclusion cannot be expanded into a precise covering condition from this evidence alone. Compactness is retained as stated; the excerpt does not explicitly specify separation axioms or exclude empty or singleton spaces.

As background, the supplied excerpt reports that $[\mathrm{Pr1},\text{ Theorem }4]$ proves that a left bounded subgroup $H\subset S(X)$ is distal when it acts transitively on $X$. This is a reported source claim, with the same unresolved terminology and an incomplete bibliographic reference.

Short English statement

For a compact space, suppose a subgroup acting transitively satisfies the stated distal condition: each distinct pair stays uniformly out of some neighborhood of every proposed common limit point under all group elements. Must that subgroup be left bounded? The supplied excerpt does not define $S(X)$ or left boundedness, so those meanings require source clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800731$, source code catalogue entry, title Open Problems in Topological Algebra --- Problem 5 (Protasov). The registry labels the problem open; that label is supplied metadata, not an independently verified status assessment.
• Banakh and Choban, Some Open Problems in Topological Algebra (2012), Problem 5. Supplied source URL: https://arxiv.org/abs/1202.1619. The record lists Banakh and Choban as proposers and gives Protasov as attribution/context.
• Supplied provenance: accessed 2026-10-16; extraction method arxiv_latex. These are supplied provenance details, not a claim of independent source consultation.
• The excerpt cites $[\mathrm{Pr1},\text{ Theorem }4]$ for the reported implication from transitive left boundedness to distality. No full bibliographic entry for $[\mathrm{Pr1}]$ is supplied.