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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