Open and Closed Ball
Let
be an arbitrary metric space. The open ball centered at of radius is the set and the closed ball centered at of radius is the set e.g. If is any set and is the discrete metric then
Interior
Let
be a metric space, and . is called an interior point of if there is such that . The set of all interior points of is called the interior of , denoted by or .
Definition
Def Exterior Let
be a metric space, and . is called an exterior point of if there is such that . The set of all exterior points of is called the exterior of , denoted by .
Definition
Def Boundary Let
be a metric space, and . is called a boundary point of if every ball contains points of and points of . The set of all boundary points of is called the boundary of , denoted by .
Proposition
Prop By definition we can see immediately that
and . and . - int(A),
and are mutually disjoint and . - If
, then and .
Proposition
Prop Let
be a metric space, and . Then
. . .
Corollary
Corollary Let
be a normed vector space and let be the induced metric. Then,
. . .