# Proof that the budget set is closed [closed]

Mas-Colell, Whinston and Green's Microeconomic Theory affirms the budget set is closed. I would like to know why.

The definition of open set in the book is the following:

## closed as off-topic by EnergyNumbers, Giskard, Adam Bailey, Kitsune Cavalry♦Dec 29 '18 at 16:02

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• What exactly are you asking? Are you asking what 'closed set' means? – Giskard Dec 21 '18 at 17:10

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).
Hint: a set is closed if its complement is open, and if a bundle $$x$$ is strictly unaffordable, $$p \cdot x > w$$, every nearby bundle is also unaffordable.