In chapter 1 section 3 of Hartshorne's Algebraic Geometry, the following proposition is given:
Let $U_i \subset \mathbb P^n$ be the open set defined by the equation $x_i \neq 0$. Then the mapping $\phi_i:U_i\rightarrow \mathbb{A}^n$ is an isomorphism of varieties, where $$ \phi(a_0,\ldots,a_n) = (a_0/a_i,\ldots,a_n/a_i) $$ with the term $a_i/a_i$ omitted.
I am confused because to suggest $U_i$ is a variety is to suggest that $\mathbb{P}^n$ is not connected, as this would make $U_i$ clopen. Am I misunderstanding the proposition, or is something else going on here that I am missing?
After reflection I realized that I forgot that an isomorphism of varieties extends to quasi-varieties - i.e. open subsets of varieties. So the statement I took issue with doesn't necessitate that $U_i$ be closed in it's ambient projective space.