TITLE: Are "axioms" in topology theory really axioms?
QUESTION [5 upvotes]: 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!

REPLY [9 votes]: The definition of axioms as "those statements that we assume to be true, instead of proving to be true" is vague enough to cover a multitude of interpretation.
The classical one considers an axiom to be an absolute mathematical truth that we're convinced of for reasons other than having seen proof. This was how axioms were viewed from Euclid's time until somewhen in the middle 19th century. Many works on mathematics for a lay audience still present it as the only one.
However, modern mathematics view it differently. According to it, axioms are simply statements that we "assume to be true" for the purpose of the argument at hand, and then worry about whether they are true later (if at all). In that sense, defining as an axiom that (say) "for any two different points there is an open set that contains one but not the other" is not meant to mean to assert a universal truth. There may well be spaces for which this is not true. Assuming it as an axiom simply means that spaces that don't satisfy it are not what we're talking about for the moment.
Point-set topology and category theory seem to be particularly fraught with large numbers of possible axioms one is supposed to mix and match among in order to find a combination that suits what one is doing -- but the same concept really pervades all areas of modern mathematics. One typically specifies axioms for some subarea, which is just a customary way to define "the things we're speaking about in this course", namely things that happen to satisfy those axioms.
Even at the very foundation of mathematics, the axioms of set theory are nowadays often viewed as replaceable parts that one explicitly chooses to work with. Most mathematicians would probably assert that the axioms of number theory ought to express some kind of universal truth, but the way they are treated in practice is not much different from the other fields anyway.