I was trying to prove the next statement:
If $X$ is an open set of $\mathbb{A}^2$ (here the field is $\mathbb{R}$), then $X$ is quasi-compact.
I was able to prove it using the properties of a Noetherian ring, but I found a problem I do not know how to fix.
First, I picked an open cover of $X$. Then I chose a countable collection of the cover (It was possible because of the Lindelöf property of the metric topology of $\mathbb{R}$ and the fact that Zariski topology is coarser than the metric one.). After this, the finite subcover can be obtained easily.
But apparently this property holds independently of the field chosen.
How is it possible to extract the finite subcover without using the Lindelöf argument? I was not able to prove it for an uncountable number of open sets in the cover only using the Noetherian property.