Why is an open subset of a variety also called a variety? 
Let $V$ be an algebraic set in $\Bbb{A}^n$. Any open subset $X$ of $V$ will be called a variety. 

This confused me. How come an open set will be called a variety? Isn't a variety the zero set of a bunch of polynomials? This seems like the complement of that zero set (I'm assuming we're talking about the Zariski Topology here). 
 A: Zariski closed sets are one things. They are often called affine. But open sets of Zariski closed sets are defined to be varieties in the book you are quoting.  They may or may not be affine. 
The intention being having an object defined locally.
A: I asked myself the very same question while reading W. Fulton's Book
'Algebraic Curves', the definitions of which I am referring to in the
following (Chapter 6, Section 2).
By definition, if $V$ is a nonempty irreducible algebraic set, then an open
subset of $V$ is called a variety, where 'open' refers to the Zariski
topology.
This definition includes $V$ itself, since $\mathbb{A}^n$ is an open subset of
itself (by definition of a topology), such that $V = V \cap \mathbb{A}^n$ is
open in the topology of $V$ induced by that of $\mathbb{A}^n$. Hence, $V$ is
an open subset of itself, and is thus a variety according to the topological
definition above.
In other words, the algebraic definition of a variety (zero locus of a prime
ideal) is more restrictive that the topological definition above. That is, the
topological definition does not contradict the algebraic definition, but it
extends it.
