TITLE: Why would the category of topological spaces be a balanced category (i.e. monic epimorphisms are isomorphisms)?
QUESTION [7 upvotes]: I've just read on this page that 

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?
Thank you for your comments!

REPLY [5 votes]: 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 $\Bbb Z \hookrightarrow \Bbb Q$).