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Math Help - The zariski topology

  1. #1
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    The zariski topology

    Hi guys,

    I have to show that the Zariski topology on an affine variety is compact. The hint is that we may use the fact that X is compact iff for every collection \{V_i\}_{i\in A} of closed sets such that \cap_{i\in A} V_i = \emptyset there exist a finite set B \subseteq A such that \cap_{i\in B}V_i = \emptyset.

    Thanks a lot in advance,

    HTale.
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  2. #2
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    Quote Originally Posted by HTale View Post
    Hi guys,

    I have to show that the Zariski topology on an affine variety is compact. The hint is that we may use the fact that X is compact iff for every collection \{V_i\}_{i\in A} of closed sets such that \cap_{i\in A} V_i = \emptyset there exist a finite set B \subseteq A such that \cap_{i\in B}V_i = \emptyset.

    Thanks a lot in advance,

    HTale.
    start with an open cover of X and take the complement to get \cap_{i\in A} V_i = X^c= \emptyset, for some closed sets V_i and a set A. then use the hint and finally take the complement to finish the proof!
    Last edited by NonCommAlg; February 11th 2009 at 04:45 AM.
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