## The Interior, Boundary and Exterior of a Set

We can define the interior, boundary and exterior of a set in using open sets.

Definition

Let be a subset of and let Then

a) is an interior point of if there is an open disc centred at which lies entirely in b) is an exterior point of if there is an open disc in centred at which lies entirely outside These definitions are illustrated below.  is an interior point of the set and is an exterior point.

The set of interior points of a set is called and the set of exterior points is called If is neither an interior or exterior point, then each open set centred at contains at least one point of and at least one point of The set of such points is called the boundary of and labelled It follows that and are disjoint and that The boundary of the set of all z satisfying is the circle Notice that if is the set the interior of is still the set of all satisfying so the boundary is still Since and are unions of open sets, they are also open, and is closed. 