# Why would the category of topological spaces be a balanced category (i.e. monic epimorphisms are isomorphisms)?

For example, $\mathsf {Set}$ (the cateogry of sets), $\mathsf {Grp}$ (the category of groups), and $\mathsf {Top}$ (the category of topological spaces) are all balanced.

(Balanced means that all the monic epimorphisms are isomorphisms).

I clearly understand for $\mathsf{Set}$ and $\mathsf{Grp}$, but isn't this wrong for $\mathsf{Top}$? For instance,

$$f:[0,1[ \longrightarrow S^1 \qquad t \longmapsto e^{2πit}$$ is continuous and bijective but is not an isomorphism in $\mathsf{Top}$. Am I missing something there?

• @RobArthan A common misconception. Epimorphisms in $\mathbf{Top}$ are surjective. – Zhen Lin Mar 5 '16 at 9:11
• @ZhenLin : In the category of Hausdorff spaces, the epimorphisms are precisely the continuous functions with dense images. For example, the inclusion map $\Bbb Q \to \Bbb R$, is a non-surjective epimorphism. (See here). – Watson Mar 5 '16 at 9:21
As it was pointed out in the comments (by Pedro Sánchez Terraf and Rob Arthan), the PlanetMath page is wrong. It is not true that every monic epimorphism in $\sf Top$ is an isomorphism.
Other examples of such morphisms can be found in the category of Hausdorff spaces $\sf Haus$ (looking at the inclusion $\Bbb Q \hookrightarrow \Bbb R$) or in $\sf Ring$ (looking at $\Bbb Z \hookrightarrow \Bbb Q$).