connectedness of A={$(x,y):x \in \mathbb{Q}\,$ or$ \, y \in \mathbb{Q}$}

Question

i have the following problem about connectedness, prove that in the Euclidean plane

A={$(x,y):x \in \mathbb{Q}\,$ or$ \, y \in \mathbb{Q}$} is connected

I have tried in several ways, but it causes me a problem, that some of the coordinates have to be rational

Answer 1

Observe that $$A = \{(x,y) \in \mathbb R^2 :\, (x,y) \notin (\mathbb R \setminus \mathbb Q)^2\} = \mathbb R^2 \setminus (\mathbb R \setminus \mathbb Q)^2,$$ and in fact, you can prove something more general:

If $X$ and $Y$ are connected topological spaces, $A$ is a proper subset of $X$ and $B$ is a proper subset of $Y$, then $(X \times Y) \setminus (A \times B)$ is connected.