How to check if a set is compact?

Question

I am struggling in finding a way to check if a set is compact. I know by definition that the set is compact if it is closed and bounded, but what about practice?

Especially if I have something like that:

$$\{(x,y)\mid x^2+y^2 < 2\}$$

Do you know how to approach the problem?

Answer 1

Sure. That specific set is not compact since it is not a closed set: $\lim_{n\to\infty}\left(\sqrt{2}-\frac1n,0\right)=(\sqrt{2},0)$, which does not belong to your set, whereas each $\left(\sqrt{2}-\frac1n,0\right)$ does.