The Stacks project

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. $\square$