Is $\mathbb{Q}$ is connected with respect to the co – countable topology ?
My attempt :Suppose A and B are disjoint nonempty open sets such that $\mathbb{Q}=A\cup B$, then both A and B are countable. But $B \subset A^c$ and $A^c$ is ucountable which is a contradiction. So $\mathbb{Q}$ is connected.
Is its true ?
Best Answer
Since $\mathbb Q$ is countable, the cocountable topology in $\mathbb Q$ is the discrete topology. And no discrete space with more than one point is connected.