Hausdorffness of $\mathbb{R}$ with cocountable topology

Question

I know that $(\mathbb{R}, \tau_{cocountable})$ is a $T_1$ space, and that it is not $T_2$. It is easy for me to prove that it is $T_1$ (let $x, y \in \mathbb{R}$, take $\mathbb{R}\backslash \{x\}$ and $\mathbb{R}\backslash \{y\}$ as the open sets). However, I can't come up with a solid proof for the fact that this space is not $T_2$. Could someone help me with this?

Answer 1

You need only to see how open sets look like. Let $U$ be a nonempty open set $\tau_{\text{co}}$; then $$U=\mathbb R\setminus(\text{countable set})$$

Now, let $x,y\in\mathbb R$, then you can not find two disjoint nonempty open sets $U,V$such that $x\in U$ and $y\in V$