Mas-Colell, Whinston and Green's Microeconomic Theory affirms the budget set is closed. I would like to know why.
Loosely speaking closed means that it includes the boundary. The boundary of the budget set is formed by those goods baskets ($x$) that imply spending all available income, i.e., such that $p·x=w.$ It is clear that they belong in the budget set and, hence, it is closed. (To be strict, we'd also need to check that points such as $ \vec 0$ also belong in the budget set. That's easy and I leave it for you to prove).