7.2 Sets and Spaces
391
space or complete for short. The complete metric space, therefore, refers to in the
metric space that a Cauchy sequence and a convergent sequence are equivalent.
If all of the ordered coordinates (x 1 , x 2 , · · · , x n ) composed of n real numbers
is a set X , and the distance between any two points P(x 1 , x 2 , · · · , x n ) and
Q(y 1 , y 2 , · · · , y n ) in X can be expressed as
ρ(P, Q) =
n
i=1
(x i − y i ) 2
then X is called the n-dimensional space, it is denoted by R
n . Every ordered coordinates (x 1 , x 2 , · · · , x n ) in R
n is called an n-dimensional point. The distance of this
definition is often called the n-dimensional Euclidean distance.
Let E be a point set in R
n , if there is a positive constant k, for all n-dimensional
points x = (x 1 , x 2 , · · · , x n ) in E, there is |x i | k(i = 1, 2, · · · , n), then E is called
the bounded set.
Let (X, ρ) be a metric space, x 0 ∈ X , r > 0 is a real number, then three types of
point sets can be defined
N (x 0 , r ) = { x ∈ X |ρ(x 0 , x) < r } (open ball)
(7.2.1)
B(x 0 , r ) = { x ∈ X |ρ(x 0 , x) r } (closed ball)
(7.2.2)
S(x 0 , r ) = { x ∈ X |ρ(x 0 , x) = r } (sphere or spherical surface)
(7.2.3)
In the above three types of point sets, x 0 is called the center of a ball, center
of sphere or spherical center, r is called the radius. The open ball N (x 0 , δ) of
radius δ is called the δ neighborhood of x 0 , it is called the neighborhood for short.
δ neighborhood of x 0 refers to that X contains arbitrary a subset of δ neighborhood
of x 0 .
From the above definition the following relation can be obtained directly
S(x 0 , r ) = B(x 0 , r ) − N (x 0 , r )
(7.2.4)
Let E be a point set in (X, ρ), x is a fixed point in (X, ρ). If there is a neighborhood
N (x, δ), such that N (x, δ) ⊆ E, then x is called the inner point, interior point or
internal point of E. If x ∈ E
C
= (X, ρ) − E and there is a neighborhood N (x, δ),
such that N (x, δ) ⊆ E
C , then x is called the exterior point or external point of E.
If x ∈ (X, ρ) is neither a interior point of E nor an exterior point of E, then x is called
a boundary point or frontier point of E. All the boundary points of E are called
the border or boundary of E. If for arbitrary N (x, δ), there are identically infinite
points belonging to E, then x is called a point of accumulation, accumulation
point or cluster point of E. Clearly the interior point of E must be an accumulation
point, and must belong to E, but an accumulation point of E does not necessarily
391
space or complete for short. The complete metric space, therefore, refers to in the
metric space that a Cauchy sequence and a convergent sequence are equivalent.
If all of the ordered coordinates (x 1 , x 2 , · · · , x n ) composed of n real numbers
is a set X , and the distance between any two points P(x 1 , x 2 , · · · , x n ) and
Q(y 1 , y 2 , · · · , y n ) in X can be expressed as
ρ(P, Q) =
n
i=1
(x i − y i ) 2
then X is called the n-dimensional space, it is denoted by R
n . Every ordered coordinates (x 1 , x 2 , · · · , x n ) in R
n is called an n-dimensional point. The distance of this
definition is often called the n-dimensional Euclidean distance.
Let E be a point set in R
n , if there is a positive constant k, for all n-dimensional
points x = (x 1 , x 2 , · · · , x n ) in E, there is |x i | k(i = 1, 2, · · · , n), then E is called
the bounded set.
Let (X, ρ) be a metric space, x 0 ∈ X , r > 0 is a real number, then three types of
point sets can be defined
N (x 0 , r ) = { x ∈ X |ρ(x 0 , x) < r } (open ball)
(7.2.1)
B(x 0 , r ) = { x ∈ X |ρ(x 0 , x) r } (closed ball)
(7.2.2)
S(x 0 , r ) = { x ∈ X |ρ(x 0 , x) = r } (sphere or spherical surface)
(7.2.3)
In the above three types of point sets, x 0 is called the center of a ball, center
of sphere or spherical center, r is called the radius. The open ball N (x 0 , δ) of
radius δ is called the δ neighborhood of x 0 , it is called the neighborhood for short.
δ neighborhood of x 0 refers to that X contains arbitrary a subset of δ neighborhood
of x 0 .
From the above definition the following relation can be obtained directly
S(x 0 , r ) = B(x 0 , r ) − N (x 0 , r )
(7.2.4)
Let E be a point set in (X, ρ), x is a fixed point in (X, ρ). If there is a neighborhood
N (x, δ), such that N (x, δ) ⊆ E, then x is called the inner point, interior point or
internal point of E. If x ∈ E
C
= (X, ρ) − E and there is a neighborhood N (x, δ),
such that N (x, δ) ⊆ E
C , then x is called the exterior point or external point of E.
If x ∈ (X, ρ) is neither a interior point of E nor an exterior point of E, then x is called
a boundary point or frontier point of E. All the boundary points of E are called
the border or boundary of E. If for arbitrary N (x, δ), there are identically infinite
points belonging to E, then x is called a point of accumulation, accumulation
point or cluster point of E. Clearly the interior point of E must be an accumulation
point, and must belong to E, but an accumulation point of E does not necessarily