How to prove that following set is not locally compact.

Question

Given set is $$X = \{(x, y)\in \mathbb{R}^2\ : \ \ x, y \ \text{irrational}\}$$ How to show that $X$ is not locally compact?

I know that a subset of $\mathbb{R}^2$ is locally compact if every point $x$ of $X$ has a compact neighborhood. How to think?

Answer 1

Assume K is compact and (0,0) subset open U subset K.
The first projection P of K should be compact, but isn't.
This is because a part of P is an interval of irrationals.