Cocountable Topology is not Hausdorff.

This question has been answered already, this is an attempt to rephrase it.

Let $(X,\tau)$ be a cocountable topology with $X$ uncountable.

Show that $(X, \tau)$ is not Hausdorff.

Let $r\not=s$, and $r\in U$ and $s \in V$, where $U, V$ are open and not empty.

Assume $U \cap V = \emptyset$.

$X\setminus (U \cap V) =$

$(X\setminus U)\cup (X\setminus V)$.

RHS is a union of countable sets, hence countable.

LHS is $X\setminus\emptyset=X$ uncountable, a contradiction.

Perhaps nitpicking :

$\emptyset$ is open and an element of the topological space, but $X\setminus\emptyset$ is not countable.

How do you go about this?

Solution 1:

The complement of any non-empty open set is countable. Since $r \in U$ and $ s\in V$ your open sets $U$ and $V$ are non-empty. So your argument works fine.