4
$\begingroup$

Let $X\subseteq \mathbb{P}^n(\mathbf{C})$ be a quasi-projective variety.

Q: Is $X$ necessarily quasi-compact in the Zariski topology (if yes then how to prove it)?

$\endgroup$
0

1 Answer 1

7
$\begingroup$

Say, any subset of a Noetherian topological space is quasi-compact with respect to the induced topology.

$\endgroup$
5
  • $\begingroup$ yes you're right, that was a dumb question $\endgroup$ Jan 26, 2014 at 22:11
  • 1
    $\begingroup$ @HugoChapdelaine no problem, happens to all of us $\endgroup$ Jan 26, 2014 at 22:28
  • $\begingroup$ nevertheless, your solution gives the right point of view. $\endgroup$ Jan 26, 2014 at 22:40
  • $\begingroup$ In fact Noetherian is equivalent to "hereditarily compact." $\endgroup$ Jan 26, 2014 at 22:59
  • $\begingroup$ @QiaochuYuan sure, that's actually a criterion. Thank you for the term though, have never heard of this one. $\endgroup$ Jan 26, 2014 at 23:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.