I want to see that in the category of topological spaces (with continuous maps) the epimorphisms (they are the continuous surjective maps) don't coincide with split epimorphisms, retractions and I need a counterexample. I got a characterization in link (page 34, 5.8. examples, (3)), namely the retractions in $\textbf{Top}$ are (up to homeomorphism) exactly the topological retractions. So a map, which is surjective and continuous, but it is not a topological retraction. Is it a counterexample?
2026-05-15 22:53:13.1778885593
The epimorphisms do not coincide with retractions in category $\textbf{Top}$.
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Consider the simplest example.
Take a set $X$ with at least two elements. Put the discrete topology on $X$ and call the space $X_1$. Put the trivial topology (or any topology other than the discrete topology) on $X$ and call the space $X_2$.
Take the identity map (of sets) $i : X_1 \rightarrow X_2$. This is continuous and surjective, and hence an epimorphism in Top.
It can't be a split epimorphism for otherwise, it would have a continuous right inverse, and bijectivity at the level of sets would imply that the right inverse would also be a left inverse, so that $i$ is an isomorphism in Top, that is a homeomorphism, which is a clear contradiction.