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Specifically, how to show that an affine variety over complex number is never compact in Euclidean topology unless it is a single point. I got a hint on this qiestion: Given an affine variety X, show that the image of X under the projection map onto the first coordinate is either a point or an open subset (in the Zariski topology).

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    After proving what's asked for in the hint, do you see how to proceed? (Consider the images of X under all of the coordinate projections. What happens if X is compact?)2011-02-23
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    In fact, I have no idea how to prove the image of X under the projection map onto the 1st coordinate is an open subset in Zariski topology.2011-02-23
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    Does someone help me?2011-02-23

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