Connected sets and irrational numbers

Question

Prove that the set $C$ of points in $\mathbb{R}^2$ such that at least a coordinate is irrational is connected set.

Ok, I know that the set of Irrational numbers isn't connected but how that helps me?

I tought in:

Let $A \subset C$ a nonempty a clopen, then we need to show that $A=C$, ok, how I can continue?

Why don't you just try to show that two points in $C$ are connected by a continuous path in $C$? — bof, Jun 24 '19 at 17:58
I am absolutely sure that I am missing several other duplicates. — Asaf Karagila, Jun 24 '19 at 17:58

score 0 · Answer 1 · answered Jun 24 '19 at 17:55

0

Hint: show it is path-connected using paths consisting of horizontal and vertical segments.

answered Jun 24 '19 at 17:55

Robert Israel

1 Answers1