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306 Advanced Linear Algebra
5 ¬ ²% Á %³
or equivalently,
5 ¬ % )²%Á ³
In this case, is called the limit of the sequence % ² . ³
%
If 4 is a metric space and is a subset of 4 , by a sequence in , we mean a
:
:
sequence whose terms all lie in : . We next characterize closed sets and
therefore also open sets, using convergence.
Theorem 12.2 Let 4 be a metric space. A subset : 4 is closed if and only if
:
whenever ²% ³ is a sequence in and ²% ³ ¦ % , then % : . In loose terms, a
subset is closed if it is closed under the taking of sequential limits.
:
Proof. Suppose that is closed and let ² % ³ ¦ % , where % : for all .
:
Suppose that %¤: . Then since % : and : is open, there exists an for
which %)²%Á ³: . But this implies that
)²%Á ³ q ¸% ¹ ~ J
which contradicts the fact that ²% ³ ¦ % . Hence, % : .
Conversely, suppose that is closed under the taking of limits. We show that
:
: % is open. Let : and suppose to the contrary that no open ball about is
%
contained in : . Consider the open balls )²%Á ° ³ , for all . Since none of
these balls is contained in : , for each , there is an % : q )²%Á ° ³ . It is
%
clear that ²% ³ ¦ % and so % : . But cannot be in both and : . This
:
contradiction implies that : is open. Thus, is closed.
:
The Closure of a Set
Definition Let be any subset of a metric space 4 . The closure of , denoted
:
:
by cl²:³ , is the smallest closed set containing .
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We should hasten to add that, since the entire space 4 is closed and since the
intersection of any collection of closed sets is closed exercise , the closure of
)
(
any set does exist and is the intersection of all closed sets containing . The
:
:
following definition will allow us to characterize the closure in another way.
Definition Let be a nonempty subset of a metric space 4 . An element % 4
:
is said to be a limit point , or accumulation point , of if every open ball
:
centered at meets at a point other than itself. Let us denote the set of all
:
%
%
:
limit points of by ² M : . ³
Here are some key facts concerning limit points and closures.
