The Stacks project

Lemma 110.78.1. There exists a system $G_1 \to G_2 \to G_3 \to \ldots $ of (abelian) topological groups such that $\mathop{\mathrm{colim}}\nolimits G_ n$ taken in the category of topological spaces is different from $\mathop{\mathrm{colim}}\nolimits G_ n$ taken in the category of topological groups.

Proof. See discussion above. $\square$

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