Regarding Choice in fields outside set theory. When authors say stuff like 

The equivalence of continuity and sequential continuity in metric spaces uses(/requires) some version of the axiom of choice.

Are they assuming that we are working inside a model of $\sf ZFC$ or something like that?
I'm asking this mostly because of a proof I read by Rudin of that fact, using the axiom of choice, but not saying so explicitly. If everything is left implicit, is it standard to assume that we're working with $\sf ZFC$ and/or a model of it?
 A: I'm answering your last question.
Upon more than century's evidence, it is scientifically proved that almost all correct mathematics can be formulated in ZFC, with eventually additional (explicitly quoted) axioms.
More precisely: Every mathematical theorem can be translated (at least theoretically) into a first order statement in the language $\{\in\}$, and  its mathematical proof  into a proof in first order logic, where axioms are taken from the infinite list ZFC. It might be the case that you need some extra power to do some calculation; this might be a sufficiently big universe, or that there are few elements in some set, or that some subset of the real line is Lebesgue measurable. Any of these additional assumptions (that is, anything outside ZFC) must be explicitely quoted somewhere in your proof. A theorem proved using assumptions outside ZFC is only conditionally proved (similarly as when you prove something under the assumption of the Riemann Hypothesis).
On the contrary, you do not need to ask for permission before using anything on the list (and AC is listed among ZFC axioms).
