Let $T=\{ U\subseteq X:X\setminus U\textrm{ is countable}\}\cup \{\emptyset\}$
Then this is known as co-countable topology.
Clearly,real line with co-countable topology is not Hausdorff.
For: If T is Hausdorff there exist two disjoint open sets that seperate the any pair of points;say G and H. Then $X\setminus (A \cap B)=X$. So the LHS is countable, but RHS is uncountable, which is a contradiction.
Can I use the same argument to prove the result if X is the complex plane or the Euclidean space?