Path connectedness of a subset of $R^2$

Question

Let us consider the following subset of $\mathbb{R}^2$.

$\cup_{ n=1}^{\infty}\{(x, y) \in \mathbb{R}^2\ | \ x = ny\} \subset \mathbb{R}^2$

How to show that above set is path connected?

Answer 1

Let $p_1=(ny,y)$ and $p_2=(my',y')$ be two points, and take the path $$ \gamma(t)=\begin{cases}(1-2t)(ny,y),&0\leq t\leq 1/2\\ (2t-1)(my',y'),& 1/2\leq t\leq 1 \end{cases} $$ I leave to you to check that the path is continuous in time.

Answer 2

Pick two point in your set. If the two points are on the same line , you connect them with the segment of that line joining them. If they are on two different lines passing through the origin,start with one the points, go through the origin with one segment and then go to the other point through the other line segment. Thus you can connect any two points with a path.