The Boundary of a Set
Suppose we have a subsetof a set
The boundary ofis labelledand is the difference between the closure of the set and the interior of the setThe setmay be open or closed - the boundary is the same and does not need to be a part of
Sincewe can also write
The boundary of a setconsists of those pointsfor which every open set containing contains points inbesidesand points inbesidesIfthen thatis a limit point ofsoandis a limit point ofsohence
Suppose thatthencan't be an interior point ofsince if there would be an open ballwithSimilarly, x can't be an interior point of X-A.
Henceandsois a boundary point ofandhence