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Given a topological ring $R$ and an arbitrary (thus not necessarily surjective) epimorphism $q: R \to S$ of underlying rings is there a finest topology on $S$ such that 1) $S$ is a topological ring and 2) $q: R \to S$ is continuous?

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    $\begingroup$ Do you mean onto or epimorphism in the category sense? $\endgroup$
    – Benjamin Steinberg
    Apr 19, 2022 at 20:40
  • $\begingroup$ Yes. Not necessarily surjective. I will edit the question to reflect this. $\endgroup$
    – user46484
    Apr 19, 2022 at 21:12
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    $\begingroup$ The set $\mathscr T$ of all topologies on $S$ making the ring operations and $q$ continuous is not empty as it contains $\{\emptyset,S\}$. Isn't the least upper bound (i.e., the initial topology on $S$ for the mappings $id:X\to (X,\tau)$ with $\tau\in\mathscr T$) the finest ring topology on $S$ making $q$ continuous? $\endgroup$
    – Jochen Wengenroth
    Apr 20, 2022 at 12:22

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