Lemma 5.30.11. Let $R$ be a topological ring. The category of topological modules over $R$ has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of $R$-modules.

Proof. It is enough to prove the existence and commutation for products and equalizers, see Categories, Lemma 4.14.11. Let $M_ i$, $i \in I$ be a collection of topological modules over $R$. Take the usual product $M = \prod M_ i$ with the product topology. Since $M \times M = \prod (M_ i \times M_ i)$ as a topological space (because products commutes with products in any category), we see that addition on $M$ is continuous. Similarly for multiplication $R \times M \to M$. Let $a, b : M \to M'$ be two homomorphisms of topological modules over $R$. 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 modules endowed with the induced topology. $\square$

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