# Why do just continuous maps are morphisms in the category of topological spaces

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.

• 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 $\mathbf{Set}$. – MJD Apr 4 '14 at 5:15