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!

  • 2
    $\begingroup$ Check ncatlab $\endgroup$ – Pedro Sánchez Terraf Mar 4 '16 at 21:43
  • $\begingroup$ @PedroSánchezTerraf : thank you for the link. Then PlanetMath seems to be wrong. $\endgroup$ – Watson Mar 4 '16 at 21:47
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    $\begingroup$ Apparently, adding corrections in PlanethMath seems difficult $\endgroup$ – Watson Mar 4 '16 at 22:19
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    $\begingroup$ @RobArthan A common misconception. Epimorphisms in $\mathbf{Top}$ are surjective. $\endgroup$ – Zhen Lin Mar 5 '16 at 9:11
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    $\begingroup$ @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). $\endgroup$ – 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 $\Bbb Z \hookrightarrow \Bbb Q$).


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