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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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up vote 4 down vote accepted

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

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