The Stacks project

Lemma 5.29.1. The category of topological spaces has colimits and the forgetful functor to sets commutes with them.

Proof. This follows from the discussion above and Categories, Lemma 4.14.12. Another proof of existence of colimits is sketched in Categories, Remark 4.25.2. It follows from the above that the forgetful functor commutes with colimits. Another way to see this is to use Categories, Lemma 4.24.5 and use that the forgetful functor has a right adjoint, namely the functor which assigns to a set the corresponding chaotic (or indiscrete) topological space. $\square$

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