Why are affine varieties except points not compact in the standard topology on $C^n$ ?

Solution 1:

Noether's normalization theorem (Mumford, Red book, page 42 ) says that if $X$ is a variety of dimension $n$ , there exists a finite surjective morphism $X\to \mathbb A^n$.
Since, in the transcendental topology over $\mathbb C$, affine space $\mathbb A^n$ is not compact for $n\geq 1$ , $X$ is not compact either.

A comment
Exciting as it definitely is, algebraic geometry has the drawback that many very intuitive facts, like the above, are difficult to justify without some fairly technical tools.
I think it is the duty of a teacher to acknowledge this explicitly in an introductory course (and maybe give a reference for the student to come back to later), rather than throw offhand remarks which might discourage a student and make him feel it is his fault that he can't find the (actually quite hard) rigorous proof.

Solution 2:

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