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
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.