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