Lemma 5.30.3. The category of topological groups has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of groups.
Proof. It is enough to prove the existence and commutation for products and equalizers, see Categories, Lemma 4.14.10. Let $G_ i$, $i \in I$ be a collection of topological groups. Take the usual product $G = \prod G_ i$ with the product topology. Since $G \times G = \prod (G_ i \times G_ i)$ as a topological space (because products commutes with products in any category), we see that multiplication on $G$ is continuous. Similarly for the inverse map. Let $a, b : G \to H$ be two homomorphisms of topological groups. 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 groups endowed with the induced topology. $\square$
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