The zariski topology

**HTale** · February 10th 2009, 01:25 PM

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.

**NonCommAlg** · February 10th 2009, 01:43 PM

Originally Posted by HTale

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!

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