The Interior, Boundary and Exterior of a Set
We can define the interior, boundary and exterior of a set in using open sets.
Letbe a subset ofand letThen
a)is an interior point ofif there is an open disc centred atwhich lies entirely in
b)is an exterior point ofif there is an open disc in centred atwhich lies entirely outside
These definitions are illustrated below.
is an interior point of the setandis an exterior point.
The set of interior points of a setis calledand the set of exterior points is called
Ifis neither an interior or exterior point, then each open set centred atcontains at least one point ofand at least one point ofThe set of such points is called the boundary of and labelledIt follows thatandare disjoint and thatThe boundary of the set of all z satisfying is the circleNotice that ifis the setthe interior ofis still the set of allsatisfyingso the boundary is still
Sinceandare unions of open sets, they are also open, andis closed.