Page 320 - Advanced Linear Algebra
P. 320
304 Advanced Linear Algebra
If 4 is a metric space under a metric , then any nonempty subset of 4 is
:
:
also a metric under the restriction of to : d : . The metric space thus
obtained is called a subspace of 4 .
Open and Closed Sets
Definition Let 4 be a metric space. Let % 4 and let be a positive real
number.
)
1 The open ball centered at , with radius , is
%
)²%Á ³ ~ ¸% 4 ²%Á %³ ¹
2 The closed ball centered at , with radius , is
)
%
)²%Á ³ ~ ¸% 4 ²%Á %³ ¹
3 The sphere centered at , with radius , is
)
%
:²%Á ³ ~ ¸% 4 ²%Á %³ ~ ¹
Definition A subset of a metric space 4 is said to be open if each point of :
:
is the center of an open ball that is contained completely in : . More
specifically, : is open if for all % : , there exists an such that
)²%Á ³ :. Note that the empty set is open. A set ; 4 is closed if its
complement ; in 4 is open.
It is easy to show that an open ball is an open set and a closed ball is a closed
set. If %4 , we refer to any open set : containing % as an open
neighborhood of . It is also easy to see that a set is open if and only if it
%
contains an open neighborhood of each of its points.
The next example shows that it is possible for a set to be both open and closed,
or neither open nor closed.
Example 12.5 In the metric space with the usual Euclidean metric, the open
s
balls are just the open intervals
)²% Á ³ ~ ²% c Á % b ³
and the closed balls are the closed intervals
)²% Á ³ ~ ´% c Á % b µ
Consider the half-open interval :~ ² Á µ , for . This set is not open, since
it contains no open ball centered at : and it is not closed, since its
complement : ~ ²cBÁ µ r ² ÁB³ is not open, since it contains no open ball
about .
