Why are affine varieties except points not compact in the standard topology on $C^n$ ? I am starting to learn algebraic geometry and in the notes I am reading there is the following remark:
" Over the complex numbers and with the strong topology we see that $A^n$ and affine varieties (except for points) are not compact. " 
(The strong topology seems to be the standard topology in this text.) 
Why is that true ? 

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.
