Wanted to explain what I think a Hausdorff is in my own words because maybe that is the root of the problem.
A Hausdorff Space is one in which for every x and y in X with x does not equal y, there exists an open set containing x and an open set containing y within x and the intersections of that is the empty set.
My first problem is I need to find a topology that isn't a Hausdorff, and that is the Cofinite topology I know from class, but I don't quite understand why.
Also, my second question is if I am given (X,T) is a metrizable space, how to I prove the topology (X,T) is a Hausdorff? I know the properties of a metrizable space, but not how they would apply to prove something is a topology, or even a Hausdorff topology.