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My question is quite simple, I would like to know why the maps (not being necessarily continuous) can't be a morphism in the category of the topological spaces, since they satisfy the properties to be a morphism (compositions are well defined, associativity and identity).

Note that the maps are the morphisms in the category of the sets, so it should be morphism also in the category of the topological spaces, since topological spaces are sets in particular.

Thanks in advance

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    $\begingroup$ Because that's not how we define the category of topological spaces. $\endgroup$ – user98602 Apr 4 '14 at 5:04
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    $\begingroup$ You could do that, but it's not interesting to do so, because then it's essentially the same as the category of sets. The objects of a category are in a certain sense incidental; it's the morphisms that are important. If you take the objects as sets with a certain structure (say, a topology) and then let the morphisms be functions that completely ignore the structure, what do you need the structure for? The theorems about the resulting category won't tell you anything about topological spaces. $\endgroup$ – MJD Apr 4 '14 at 5:07
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    $\begingroup$ Maybe you should look up the words "full functor" and "faithful functor." Not all functors are required to be full or faithful. This is particularly true of the so-called "forgetful functors." Under your proposed definition, category of sets and category of spaces would become equivalent categories if you think it through. $\endgroup$ – user36931 Apr 4 '14 at 5:08
  • $\begingroup$ @MJD It's true, thanks $\endgroup$ – user42912 Apr 4 '14 at 5:08
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    $\begingroup$ You might consider this simpler example of the same type: let the objects of $\mathbf{Set}^\ast$ be pointed sets, which are sets from which a single element has been somehow distinguished. Usually we take the morphisms of $\mathbf{Set}^\ast$ to be functions that map distinguished points to distinguished points, but we can follow your idea and take them to be ordinary functions instead. Now consider in what way this category is different from the usual $\mathbf{Set}$. $\endgroup$ – MJD Apr 4 '14 at 5:15
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In general, you want maps which preserve the relevant structure of the objects in your category. So, in the category of groups, you want your morphisms to preserve the group structure, i.e. group homomorphisms. In the category of vector spaces over a given field, you want linear transformations. So, in the category of topological spaces, you want continuous maps.

Note that in each of these examples, the morphisms are still morphisms if we forget to the underlying category of sets.

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