Is $(\mathbb Q \times \mathbb Q)$ connected?
I am assuming it isn't because $\mathbb Q$ is disconnected. There is no interval that doesn't contain infinitely many rationals and irrationals.
But how do I show $\mathbb Q^2$ isn't connected? Is there a simple counterexample I can use to show that it isn't? What would the counterexample look like?
Best Answer
Suppose $\mathbb{Q}\times \mathbb{Q}$ was connected, then by calling $\pi:\mathbb{Q}\times \mathbb{Q} \to \mathbb{Q}$ one the projections, you would get that $\mathbb{Q}$ is connected, since it would be a continuous image of a connected space.