1. The problem statement, all variables and given/known data Let X be a set equipped with a topology tau1, and let tau2 be the cocountable topology in which a set V in X is an open set if V is empty or X - V has only finitely or countably many elements. Consider the topology tau consisting of all sets W in X such that for each point p in W there exist subsets A and B of X containing p such that A is open in tau1, B is open in tau2, and the intersection of A and B is a subset of W. If (X, tau1) has a countable local base at a point p in X, then under what conditions does (X, tau) have a countable local base at p? 2. Relevant equations 3. The attempt at a solution I am truly at a total loss on how to do this. I've been messing around with this problem for a long time and am just not seeing the solutions. At this point, I honestly don't have any work worth showing. I don't know if I'm just having a brain meltdown right now or if this is an unexpectedly hard problem.