Shafarevich in the book "Basic Algebraic Geometry I" gives the following definition of a quasi-projective variety:
A quasi-projective variety is an open subset (respect to the induced Zariski topology) of a closed projective set.
Then he says that a closed affine set is a quasi-projective variety, but i disagree with this statement. Here I'll try to show why:
We know that $\mathbb P^n_k=U_0\cup\ldots\cup U_n$, where $U_i$ is open and $U_i\cong\mathbb A_k^n$. If $X$ is a closed affine set and $\overline X$ is its projective closure (note that $\overline X$ is a closed subset of $\mathbb P^n_k$ and it contains a "copy" of $X$), one can show that $X$ is homeomorphic to $\overline X\cap U_i$ which is open in $\overline X$. So technically $X$ is homeomorphic to a quasi-projective variety, but according to the previous definition it is not a quasi-projective variety because $X\not\subset\mathbb P^n_k$. Where is the mistake in my argumentation?
Clearly Shafarevich is working into the classical framework of Algebraic Geometry, so without mentioning the concept of scheme or sheaf. For this reason I'd like an answer concerning only "classical arguments". I know that this is not the most elegant way to introduce varieties, but I want to understand the abstract concepts through sucessive generalizations.