# Does quasi-projective imply quasi-compact (in the Zariski topology)?

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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## 1 Answer

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

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