Lemma 5.30.8. The category of topological rings has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of rings.
Proof. It is enough to prove the existence and commutation for products and equalizers, see Categories, Lemma 4.14.11. Let $R_ i$, $i \in I$ be a collection of topological rings. Take the usual product $R = \prod R_ i$ with the product topology. Since $R \times R = \prod (R_ i \times R_ i)$ as a topological space (because products commutes with products in any category), we see that addition and multiplication on $R$ are continuous. Let $a, b : R \to R'$ be two homomorphisms of topological rings. Then as the equalizer we can simply take the equalizer of $a$ and $b$ as maps of topological spaces, which is the same thing as the equalizer as maps of rings endowed with the induced topology. $\square$
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