So the axiom of choice is confusing to me, apperently there is a distinction between the exsistence of an element and the actual selection of an element? I'm confused as to how much the axiom of choice is needed in elementary metric space theorems? As an example, is the Axiom of Choice needed to prove the equivlent definitions of compactness for metric spaces?(Covering compact = sequentually compact = complete + totally bounded) What about the Baire Category Theorem? This PDF(http://math.berkeley.edu/~jdahl/104/104Baire.pdf [Broken])claims that the Axiom of choice is not needed to prove it in complete, totally bounded metric spaces but it appears the person still makes implict use of it. I guess I am confused in general as to when it is implicitly used in a proof.(as in the statement "select a s_n from each U_n.....".