Is compact $T_1$ topological space hausdorff?

Question

I'm in a middle of a very hard exercise which its goal is to prove that some space is hausdorff, but all I could show is that it is $T_1$. But I can also deduce that it is compact. Is that enough for a space to be hausdorff? It accured to me that $T_2$+compact implies $T_4$, so it is reasonable for such connections to be true, yet I couldn't think of a proof for that.

thanks

Answer 1

No: an infinite space with the cofinite topology is compact and $T_1$ but not Hausdorff.