With regards to vector spaces, what does it mean to be 'closed under addition?' My linear algebra book uses this term in the definition of a vector space with no prior explanation whatsoever.  It proceeds to then use this term to explain the proofs.  
Is there something painfully obvious I'm missing about this terminology and is this something I should already be familiar with?
The proof uses $u + v$ is in $V$

Consider the collection of points that literally lie on the $x$-axis or on the $y$-axis. We could still use Cartesian vector addition to add two such things together, like $(2,0)+(0,3)=(2,3)$, but we end up with a result that is not part of the original set. So this set is not closed under this kind of addition.
If addition is defined at all on a set, to be closed on that set, the result of an addition needs to still be in that set.
