# An affine space of positive dimension is not complete

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?) –  Brad Feb 23 '11 at 18:57
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. –  user7419 Feb 23 '11 at 19:11
Does someone help me? –  user7419 Feb 23 '11 at 19:56

As Plop states, the hint follows from Chevalley's theorem. However, in this context one shouldn't need to appeal to the full strength of that theorem.

In fact, Chevalley's theorem is a variation on Noether normalization (and both are variations on the Nullstellensatz --- see this MO answer), but Noether normalization is usually taught at an ealier stage than Chevalley's theorem, so you might consider using it instead. (Regard this as an alternative hint.)

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Thanks Matt E & Plop. Matt E, could you show me how to use the Noether normalization to solve my problem? –  user7445 Feb 24 '11 at 5:48
@charm: Dear Charm, Firstly, do you know the statement of Noether normalization? Regards, –  Matt E Feb 24 '11 at 5:50

The hint is a consequence of Chevalley's theorem on constructible sets: http://en.wikipedia.org/wiki/Constructible_set_%28topology%29

A non-empty Zariski-open susbset of the affine line is clearly not compact, so the image has to be a finite set. Projecting on each coordinate, you get that the variety is finite.

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