Suppose we have a subset
of a set
The boundary ofis labelled
and is the difference between the closure of the set
and the interior of the set
The setmay be open or closed - the boundary is the same and does not need to be a part of
Since
we can also write
The boundary of a setconsists of those points
for which every open set containing contains points inbesidesand points in
besides
If
then thatis a limit point ofso
andis a limit point ofso
hence
Suppose that
thencan't be an interior point ofsince if
there would be an open ball
with
Similarly, x can't be an interior point of X-A.
Hence
and
sois a boundary point ofandhence
Hence