Page 321 - Advanced Linear Algebra
P. 321
Metric Spaces 305
Observe also that the empty set is both open and closed, as is the entire space .
s
(Although we will not do so, it is possible to show that these are the only two
s
sets that are both open and closed in .³
It is not our intention to enter into a detailed discussion of open and closed sets,
the subject of which belongs to the branch of mathematics known as topology .
In order to put these concepts in perspective, however, we have the following
result, whose proof is left to the reader.
E
Theorem 12.1 The collection of all open subsets of a metric space 4 has the
following properties:
1 J E ) , 4 E
2 If , :; E ) then : q ; E
)
3 If ¸: 2¹ is any collection of open sets, then 2 : E .
These three properties form the basis for an axiom system that is designed to
generalize notions such as convergence and continuity and leads to the
following definition.
Definition Let be a nonempty set. A collection of subsets of is called a
?
E
?
topology for if it has the following properties:
?
1) J E Á ? E
)
2 If :Á ; E then : q ; E
)
E
3 If ¸: 2¹ is any collection of sets in , then : E .
2
We refer to subsets in as open sets and the pair ²?Á ³ as a topological
E
E
space.
According to Theorem 12.1, the open sets as we defined them earlier in a
)
(
metric space 4 form a topology for 4 , called the topology induced by the
metric.
Topological spaces are the most general setting in which we can define concepts
such as convergence and continuity, which is why these concepts are called
topological concepts. However, since the topologies with which we will be
dealing are induced by a metric, we will generally phrase the definitions of the
topological properties that we will need directly in terms of the metric.
Convergence in a Metric Space
Convergence of sequences in a metric space is defined as follows.
Definition A sequence ²% ³ in a metric space 4 converges to % 4 , written
²% ³ ¦ %, if
lim ²% Á %³ ~
¦B
Equivalently, ²% ³ ¦ % if for any , there exists an 5 such that
