I am a beginner in topology and am struggling in writing down examples of topological spaces besides from the most trivial ones and some common metric spaces. I need a rather simple example of a topological space that satisfies the first axiom of countability but where the second one does not hold. I thought about the simple Euclidean metric in $R^1$ but it seems to me that this does have a countable base and therefore must satisfy the FAC. What could be an example? 

Thank you.