the definition of the intersection of a collection of sets:
the collection of element x such that x belongs to U_i for all i in I.
so we don't need AC to prove this definition.
Let be a collection of sets, all containing a common element for each .
Do we need the axiom of choice to show that ?
The reason I ask is because wikipedia says that we need the axiom of choice to prove the Baire Category Theorem and in the proof it gives on the page, I can't see where this would be if it wasn't in the above.
Well if you just talk about one element or one set from an infinite collection, there is no problem, you can choose one and "fix" it. Problems arise when you want to fix an infinite amount of things at the same time; in such cases the axiom of choice can become necessary.