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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)?

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

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  • $\begingroup$ yes you're right, that was a dumb question $\endgroup$ – Hugo Chapdelaine Jan 26 '14 at 22:11
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    $\begingroup$ @HugoChapdelaine no problem, happens to all of us $\endgroup$ – Anton Fonarev Jan 26 '14 at 22:28
  • $\begingroup$ nevertheless, your solution gives the right point of view. $\endgroup$ – Hugo Chapdelaine Jan 26 '14 at 22:40
  • $\begingroup$ In fact Noetherian is equivalent to "hereditarily compact." $\endgroup$ – Qiaochu Yuan Jan 26 '14 at 22:59
  • $\begingroup$ @QiaochuYuan sure, that's actually a criterion. Thank you for the term though, have never heard of this one. $\endgroup$ – Anton Fonarev Jan 26 '14 at 23:03

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