[SOLVED] very basic topology questions 1. The problem statement, all variables and given/known data Let X be a set and T be the collection of X and all finite subsets of X. When is T a topology? Let T' be the collection of X and all countable subsets of X, when is T' a topology? 3. The attempt at a solution it's clear the empty set and X are in T if two finite subsets united, the new set is also a finite subset the intersection between two finite subsets is again finite. The only hole I can find is when I unite an infinite amount of finite substes of X, but what restriction does that give on X? Surely, if X is finite T is a topology... but surely I can say a bit more than that? I use the same reasoning for the countable version of the question, and I find that if X is countable T' is a topology... but again, I'm left wondering if there is nothing more to say.