Edited: Why do theorems in one area of Maths often become axioms in another area?

Question

I was reading a theorem from "Beals -- Analysis an Introduction", about how given two metric spaces $(X,d)$ and $(Y,p)$. A function $f: X \to Y$ is continuous on $X$ if and only if for all open sets $O$ in $Y$, the pre-images $f^{-1}(O)$ are open in X.

This theorem is taken as an axiom in more axiomatic approaches to Topology. I have heard that this approach is a common theme in Maths. Why is that? I realize it saves time by not having to re-invent the wheel, and hides the clutter behind the curtains. But I have been thinking a lot about how a Logician would reduce statements in Higher Maths to first-principles so perhaps this is an admission that this example is not the best to pose this question (as this proof is of the type "the next step is always what is the only thing you can do" scenario)

But I really want to ask anyone who is reading how can one be sure that subtle conditions that are required in the proof of a statement in the "niche context" be exactly matched in the later works if this Theorem is adopted as an Axiom? Do Logicians have to worry about this? Like if the Law of Excluded Middle is not applicable to the Theorem we are now adopting as an Axiom, then we have to discard proof by contradiction from our Mathematical toolkit.

Other rough thoughts:
Open sets forms the basis of Topology. And as some sets can be both open and closed and some can be neither open nor closed. Does it also mean that L.E.M is not applicable anywhere in Topology and hence proof by contradiction a banned toolkit?

P.S sorry to anyone who saw this before it was meant to be posted. The question was only half written and didn't make any sense. I am new to this website.

Answer 1

1. Let's get out of the way the easy bit. "As some sets can be both open and closed and some can be neither open nor closed. Does it also mean that L.E.M is not applicable anywhere in Topology and hence proof by contradiction a banned toolkit?" No. "Open" -- in topology -- doesn't mean "not closed", nor does "closed" mean "not open". Check out the official definitions of these two notions in your favoured topology textbook. [You might complain, I suppose, that the use of jargon is somewhat unhelpful, given that in ordinary contexts, "open" and "closed" are contradictory. But you just have to be be alert for cases like this, where mathematicians borrow vivid ordinary words for a not unrelated technical use, although some of the implications of the ordinary words are lost in the mathematical context.]
2. As to the more general point about the phenomenon of the result of a theorem here being treated as an axiom there. There are various sorts of case here. But one (not untypical) sort can be illustrated like this. We give, say, axioms for the real numbers as a complete ordered field -- and show that, on these assumptions, lots of consequences follow. However, we wonder, are there any complete ordered fields? (Indeed, are those axioms even consistent??) We then show that, yes, in a set theoretic framework, we can construct a complete ordered field -- we define Dedekind cuts on the rationals, say, and then prove lots of theorems to the effect that these cuts form a complete ordered field. Good: the theorems show that our field axioms are indeed true of some set-theoretic structure.
3. But having done that, for many purposes, we can now forget about the construction -- the theorems comfort us by telling us that we are not talking inconsistent nonsense in adopting the axioms of a complete ordered field. But once we know there are such fields, for many purposes we just don't care which complete ordered field we are talking about: we can and do forget about the underpinning. We abstract from any "concrete" implementation of the structure (and hence forget about the theorems which tell about one such concrete implementation), and now just think more generally: given these propositions about complete ordered fields, treated as axioms and not worrying about particular implementations, what follows about any structure satisfying these axioms?

Answer 2

I assume that you refer to that we first define continuity of $f : \mathbb R \to \mathbb R$ by $\lim_{x \to a} f(x) = f(a)$ for all $a \in \mathbb R,$ generalize this to other metric domains $X$ and target spaces $Y,$ then show in an theorem that $f$ is continuous if and only if $f^{-1}(V)$ is open in $X$ for every $V$ that is open in $Y$, and finally take this property as a definition of continuity for maps between general topological spaces.

My answer to your question is that this is a big part of generalization. Find common properties and use these as axioms in a new area.

For example, square matrices with non-zero determinant can be composed and inverted. Also translations can be composed and inverted. Several other things can be composed and inverted (reverted). The common operations then becomes the foundation of group theory.