finding connected components

Question

I want to find the number of connected components of $\mathbb{R}^2\backslash\mathbb{Q}^2$. My approach is since $\mathbb{Q}^2$ is a countable set. Then its compliment that is $\mathbb{R}^2\backslash\mathbb{Q}^2$ is path connected and thus connected. So does that mean it has one connected component? And if we replace $\mathbb{Q}^2$ by $(\mathbb{Q}^c)^2$, then how many connected components $\mathbb{R}^2\backslash(\mathbb{Q}^c)^2$ will have ? Assume the study under usual topology.

Answer 1

It seems that you already know that $\mathbb{R}^n\backslash A$ is connected whenever $n\geq 2$ and $A$ is countable. Thus clearly $\mathbb{R}^2\backslash\mathbb{Q}^2$ is connected. Therefore it has exactly one connected component.

This reasoning obviously doesn't apply to $(\mathbb{Q}^c)^2$ because that set is not countable. And indeed, for arbitrary uncountable $A$ the space $\mathbb{R}^n\backslash A$ need not be connected, e.g. $A=\{0\}\times\mathbb{R}^{n-1}$. But

$$\mathbb{R}^2\backslash(\mathbb{Q}^c)^2=\mathbb{R}\times\mathbb{Q}\cup\mathbb{Q}\times\mathbb{R}$$

and thus given a point $(a_1,a_2)$ with, say, $a_2\in\mathbb{Q}$ we can construct first a path from $(a_1,a_2)$ to $(0,a_2)$ by $t\mapsto (ta_1,a_2)$. Note that $(ta_1,a_2)\in\mathbb{R}^2\backslash(\mathbb{Q}^c)^2$ for any $t\in\mathbb{R}$. Then we connect $(0,a_2)$ to $(0,0)$ with similar path. And finally we glue both those paths to obtain that every point in $\mathbb{R}^2\backslash(\mathbb{Q}^c)^2$ can be connected to $(0,0)$ and thus $\mathbb{R}^2\backslash(\mathbb{Q}^c)^2$ is path connected as well.