1. 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 $\displaystyle X$ is compact iff for every collection $\displaystyle \{V_i\}_{i\in A}$ of closed sets such that $\displaystyle \cap_{i\in A} V_i = \emptyset$ there exist a finite set $\displaystyle B \subseteq A$ such that $\displaystyle \cap_{i\in B}V_i = \emptyset$.

HTale.

2. 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 $\displaystyle X$ is compact iff for every collection $\displaystyle \{V_i\}_{i\in A}$ of closed sets such that $\displaystyle \cap_{i\in A} V_i = \emptyset$ there exist a finite set $\displaystyle B \subseteq A$ such that $\displaystyle \cap_{i\in B}V_i = \emptyset$.

start with an open cover of X and take the complement to get $\displaystyle \cap_{i\in A} V_i = X^c= \emptyset,$ for some closed sets $\displaystyle V_i$ and a set $\displaystyle A.$ then use the hint and finally take the complement to finish the proof!