TITLE: Hausdorff Topological Space
QUESTION [5 upvotes]: 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.
Thanks!

REPLY [3 votes]: Firstly, if you don't know what is the induced topology related to a metric space is, then you can't answer your second question so you should first review that. 
As for you definition of what a Hausdorff space is, which seems basically correct, try to visualize what is going on. Being Hausdorff means that any two distinct points can be separated by disjoint open sets. Try to draw a picture that illustrates that. As for an example of a non-Hausdorff space, you can start off by showing some very simple example of such spaces. For instance, the space $X={1,2}$ with the topology $\{X,\emptyset \}$ is not Hausdorff.