I want to check if the following set is connected, and path-connected.
A set can be path-connected only if it is connected, or not?
$$\displaystyle{D=\{x\in \mathbb{R}^2 : x_1 \text{ or } x_2\in \mathbb{Q}\}}$$
The set is equivalent to $\displaystyle{\mathbb{Q}\times\mathbb{R} \cup \mathbb{R}\times\mathbb{Q}}$.
The set of rational numbers is not connected, since it ban be written as a union of two non-empty sets: $\mathbb{Q} = ((-\infty, \sqrt{2}) \cap \mathbb{Q})\cup((\sqrt{2}, \infty)\cap\mathbb{Q})$.
From that we get that neither the set $D$ is connected, also that implies that $D$ is not path-connected, right?
D is path connected because it is a collection of verticle lines criss crossing a collection of horizonal lines.
It is like a screen with a very tiny mesh or graph paper with infinitesmial squares