1. The problem statement, all variables and given/known data (i) Let U be a topology on Z, the integers in which every infinite subset is open. Prove that U is the discrete topology. (ii) Use (i) to prove that if U is a topology on an infinite sex X in which every infinite subset is open, then U is the discrete topology on X. 2. Relevant equations None other than the definition of a topology. 3. The attempt at a solution I have solved (i) (at least I think I have). But since X is not specified to be countable I have no idea how to apply this result to the second part. A possible idea is to consider the possibilities where X is countable and uncountable seperately and set up a bijection with Z in the countable case, but I think this is over-complicating things. I am pretty stumped here.