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I am trying to think about the axiom of choice from a categorical point of view. I found out in some book that one possible formalization of this axiom is obtained by stating: "a category $\mathcal{C}$ satisfies the axiom of choice if all epics in $\mathcal{C}$ are split". In this way $\mathbf{Set}$ does satisfy the axiom of choice (and that is fine). The category of groups doesn't though! What about $\mathbf{Top}$, i.e. the category of topological spaces? Does a not split epimorphism exist in $\mathbf{Top}$?

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Most epimorphisms in $\mathbf{Top}$ do not split. For instance, let $Y$ be any nondiscrete topological space and let $X$ be the underlying set of $Y$ with the discrete topology. Then the identity map $X\to Y$ is epic but does not split (since the identity $Y\to X$ is not continuous). There are many many many other examples; I encourage you to try coming up with some on your own.

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Top indeed does not satisfy the axiom of choice. To show this it suffices to exhibit a fiber bundle without a section, and for example the unit tangent bundle of $S^2$ has this property by the hairy ball theorem. Various other kinds of examples are possible; see, for example, this blog post.

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