Open Problems in Topological Algebra — Problem 4 (Protasov)

Definitions and mathematical statement

A subset of a topological space is clopen if it is both open and closed. A space $X$ is homogeneous if, for every $x,y\in X$, there is a homeomorphism $h:X\to X$ with $h(x)=y$. The clopen-base meaning of zero-dimensional is that $X$ has a base consisting of clopen sets; the supplied excerpt does not specify whether additional separation assumptions are intended.

Source question (Protasov). Is the semigroup $S(X)$ right bounded for every zero-dimensional homogeneous space $X$?

The supplied record does not define $S(X)$, its multiplication or any relevant topology, or the predicate "right bounded." These symbols and terminology are therefore retained as unresolved source terminology; a fully specified mathematical proposition requires their definitions from the source context. No additional separation or nonemptiness assumptions are supplied.

Supplied qualification. The record reports an affirmative answer when $X$ has a topological base $\mathcal B$ such that every $B\in\mathcal B$ is clopen and any two members $B,C\in\mathcal B$ are homeomorphic with their subspace topologies. It refers to $[\mathrm{Pr1}]$ for this assertion, but does not supply that reference's bibliographic entry. This qualification is reported from the record and is not independently verified.

Short English statement

The question attributed to Protasov asks whether $S(X)$ is right bounded for every zero-dimensional homogeneous space $X$. The record reports a positive answer when $X$ has a base of pairwise homeomorphic clopen sets. Definitions of $S(X)$ and right boundedness, and the intended separation assumptions, remain to be recovered from the source.

Sources

• Local UnsolvedMath registry, permanent ID $800730$, source problem code catalogue entry, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied registry labels the problem open; this is a supplied status, not an independently verified literature assessment.
• Banakh and Choban, Some Open Problems in Topological Algebra (2012), Problem 4, attributed in the supplied record to Protasov. Supplied source URL: https://arxiv.org/abs/1202.1619. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of access or verification for this draft.
• The supplied excerpt cites $[\mathrm{Pr1}]$ for the affirmative answer under the pairwise-homeomorphic-clopen-base hypothesis. No bibliographic details for $[\mathrm{Pr1}]$ are included in the record.