1. ## Rationals, Pathwise connected

Problem:
Show that the set $\displaystyle S = \left\{ (x,y) \in \mathbb{R}^2 | x or y \in Q \right\}$ is pathwise connected.
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I can only think of a generalized path.

Since $\displaystyle 0 \in Q$, then there is a parametric path consisting of two lines conneting the x-axis and y-axis. So the parametric path is $\displaystyle \gamma (t) = (0,0)$ and the parameteric path is dependent on the values of x, y.

If x > 0, y > 0, then $\displaystyle \gamma: [0, x] \rightarrow [0, y]$
If x < 0, y < 0, then $\displaystyle \gamma: [x, 0] \rightarrow [y, 0]$

and vice versa. However, I do not know how to prove this for all points.

Thank you for reading. Any help is greatly appreciated.

2. You are almost done...

Let us show that every point of S is connected to (0,0). You proved that this is true for the points lying on either of the two axes.

Now, if (x,y) is in S, either x or y is rational. If x is rational, I guess you know how to connect (x,y) to the x-axis, and hence to (0,0). And you can do the same with the y-axis if y is rational.

At that point, we are done: if M and N are in S, you can connect M to (0,0) and then (0,0) to N.

Laurent.