Open Problems in Topological Algebra — Problem 9 (Guran)

Definitions and mathematical statement

A paratopological group is a group $G$ equipped with a topology such that multiplication $G\times G\to G$, $(x,y)\mapsto xy$, is jointly continuous. Continuity of inversion is not required. Write $e$ for the identity and $\mathcal N_G(e)$ for its neighborhoods. For subsets $A,B\subseteq G$, put $AB=\{ab:a\in A,\ b\in B\}$. Here countable includes finite.

The group $G$ is left $\omega$-bounded if \[ \forall U\in\mathcal N_G(e)\ \exists F\subseteq G\quad \left(F\text{ is countable and }G=FU\right). \] It is right $\omega$-bounded if \[ \forall U\in\mathcal N_G(e)\ \exists F\subseteq G\quad \left(F\text{ is countable and }G=UF\right). \] It is $\omega$-bounded if both properties hold. The countable set may depend on the neighborhood.

Question (Guran; supplied source, Problem 9). Does every right $\omega$-bounded paratopological group satisfy left $\omega$-boundedness? Explicitly, does the implication \[ \begin{aligned} &\bigl[\forall U\in\mathcal N_G(e)\ \exists F\subseteq G\quad (F\text{ is countable and }G=UF)\bigr]\\ &\qquad\Longrightarrow \bigl[\forall V\in\mathcal N_G(e)\ \exists H\subseteq G\quad (H\text{ is countable and }G=HV)\bigr] \end{aligned} \] hold for every paratopological group $G$?

The supplied excerpt specifies no separation axiom. Whether the cited source imposes a standing separation convention on paratopological groups remains unresolved; the intended class of groups requires confirmation.

Short English statement

Must a paratopological group in which every identity neighborhood covers the group by countably many right translates also admit a cover by countably many left translates of every identity neighborhood? The source's intended separation assumptions require confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $800735$, source code catalogue entry, titled Open Problems in Topological Algebra --- Problem 9 (Guran). The supplied record labels the problem open; this status is not independently verified.
• Banakh and Choban, Some Open Problems in Topological Algebra (2012), Problem 9, https://arxiv.org/abs/1202.1619. The supplied background attributes the problem to Guran; the registry lists Banakh and Choban in its proposer field.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims, not evidence of independent source consultation for this draft. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.