Connected subsets of $\mathbb{R}^2$

Question

Is $\mathbb{Q} \times \mathbb{Q}$ a connected subset of $\mathbb{R}^2$? When we take open spheres it will contain irrationals.. How can I do this type of problems.

Answer 1

hint: if $\mathbb Q \times \mathbb Q$ were connected in the product topology, then the projection $\pi:\mathbb Q^2 \to \mathbb Q$ would be continuous, implying that$\dots$ which is a contradiction.

Answer 2

First tip: try to use MathJax! Second tip: try to explain what did you try to do to solve the problem, and what passages are the most difficult to you.

However, consider $X=\mathbb{Q}\times\mathbb{Q}$ and the subspaces Y=$(-\infty,\sqrt{2})\times \mathbb{Q}$ and its complementary $Y^{c}=(\sqrt{2},\infty)\times\mathbb{Q}.$ They are open in X (considering the subspace topology from $\mathbb{R}^2),$ and so X is not connected.