In the context of regular languages, must the alphabet be finite?
It makes sense in some contexts in mathematics to consider strings or languages over infinite alphabets. For instance, this concept is used in the strong version of Higman's lemma. But a finite automaton requires a finite alphabet, and only finitely many symbols can actually appear in a single regular expression. So, specifically for the context of regular languages, alphabets must be finite.

It's not important whether you achieve this by declaring that "alphabet" means a finite set (as HMU and Sipser do) or whether you allow infinite alphabets and instead put the requirement of finiteness elsewhere than the definition of that word (as AU do).