As a preliminary note, I have absolutely no background in category theory. The concept was very briefly introduced in an introductory topology book I was reading as motivation for the introduction of groups, and it was left with very little description. Anyway, my question is the following - when speaking of the category of topological spaces, the book described the objects as topological spaces (as expected), but it stated that the morphisms were continuous maps. Here's where I'm confused. Is it simply convenient in further studies to define the morphisms as continuous mappings, or is it necessary to do so?
My question is much more basic than that. What I'm asking is, did the author introduce the morphisms of Top as being continuous maps because they have to be continuous, or did he introduce them as being continuous maps because when working with the morphisms of Top, the continuous maps are the important ones to study?
I guess more fundamentally, what I'm asking is would allowing the morphisms to just be any map between topological spaces be an invalid choice?
There are some pretty wild categories. You can create pretty much any category you want, as long as the axioms are respected. Whether it will be interesting or not is another question!