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In reading Hartshorne,a topological space is quasi-compact if each open cover has a finite subcover(P80).Isn't it the definition for compactness of topological spaces?Am I right?Is quasi-compactness only in use in algebraic geometry in place of compactness?Or do we have another definition for compactness in algebraic geometry?Will someone be kind enough to say something on this?Thank you very much!

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There's a disscusion: mathoverflow.net/questions/16971/compact-and-quasi-compact –  Ch Zh Aug 12 '11 at 6:50

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Hartshorne reserves "compact" for Hausdorff spaces, which many spaces in algebraic geometry fail to be. I'm not sure how prevalent this distinction is.

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In algebraic geometry many spaces are (quasi-)compact, so it sounds strange to consider $\mathbb{A}^n_{\mathbb{C}}$ a "compact" space. Therefore one prefers to use the adjective "quasi-compact". Finally, the corresponding concept in algebraic geometry of "compact space" is "proper". –  Andrea Aug 12 '11 at 7:52

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