Consider $Q_p$ and $R$ where $Q_p$ is $p-$adic numbers with $p-$adic topology. If $Q_p\to R$ is a ring homomorphism, then certainly it must be trivial as $Z\to Z$ inducing $Q\to Q$ but infinite series of p-adic element does not converge in $R$ in general.

$\textbf{Q:}$ It is natural to ask whether one can classify all continuous maps $Hom_{Cts}(Q_p,R)$ besides constant maps. It is certainly possible to get just set theoretical embedding by axiom of choice. Does $Hom_{Cts}(Q_p,R)$ have non-constant functions? I do not see an obvious choice of candidate for non-constant maps.

There are many continuous, non-constant functions; for example, the $p$-adic norm is one such (as is any positive power of the $p$-adic norm).

September 11, 2019 18:13 PM

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