Please forgive me if this question has already been asked in one form or another (in which case, it would suffice to provide the link to the corresponding post which has the answer).
My question: What would happen if we parted ways with uncountable choice? In other words, imagine working within the fragment of ZFC in which you have all of the usual axioms at your disposal (including countable choice), but you throw out the uncountable form of the axiom of choice. What sorts of problems might arise while working within this system? What are some non-trivial examples of theorems which can be "saved" (in the sense that they only required countable choice to begin with) and theorems which "break" (in the sense that their proofs explicitly depended on uncountable choice)?