Are "axioms" in topology theory really axioms? If I understand correctly, axioms are those statements that we assume to be true, instead of proving to be true.
I have seen that in topology theory, various axioms of countability and separation axioms seem to be definitions of some concepts, instead of being assumptions. So I was wondering if "axioms" in topology theory are really axioms?
If no, is this kind of naming rules common in all other branches of mathematics?
Thanks and regards!

Axioms are merely background assumptions defining the objects that are to be under consideration. If you’re studying Abelian groups, you’ll want to take commutativity of the group operation as an axiom. If you’re studying manifolds, you may want to take second countability as an axiom. For most mathematical purposes you probably want to take the axiom of choice as an assumption. What axioms you choose for your geometry depends on what kind of geometry you want. And so on. The notion that axioms are true is, as André said in the comments, very dated.
