09 Metric Spaces
We will take a break from the textbook before moving on to Chapter 3, to
look at a topic not covered in the book:
metric spaces.
Basic properties of metric spaces are covered in the readings
Open and Closed Sets
and Subspaces and Product Spaces
(but we will not spend any time on product spaces).
A metric space is a set together with a measure of distance between pairs of
points in that set. A basic example is the set of real numbers with the usual
notion of distance, where the distance between and is .
In the general definition of metric spaces, some basic properties of absolute value are
used as the defining axioms. The triangle inequality plays a prominent role.
Definition: A metric space
is a pair where is a set and is a function satisfying:
- , for all ;
- if and only if ;
- symmetry: , for all ; and
- the triangle inequality: , for all .
The function is said to be a metric on .
So, is a metric space with . becomes the metric space
with the usual notion of distance,
But there are other distance measures on that also make it into a metric space, such as
There are common metrics on other sets that you might not ordinarily think
of in terms of distance. For example on the set
of continuous real-valued functions on the interval ,
we can define a distance measure .
In any metric space, we can talk about "open" and "closed" sets. In , we
defined a set to be open if for every , there is an
such that the open interval is a contained in .
Equivalently, is open if it is a union of open intervals. We haven't
defined closed sets in , only closed intervals, but the general definition is easy:
A closed set is closed if its complement, , is open.
(Note, by the way, that it is not true that every closed set is a union of
closed intervals; it is not even true that a union of closed intervals has to be closed.)
All of these definitions extend to arbitrary metric spaces, starting with "open ball,"
which is the generalization of an open interval of the form .
Definition:
Let be a metric space. For and we define
the open ball about of radius to be the
set (When the metric is clear
from context, can be written .)
Definition:
Let be a metric space.
A subset of is said to be open in if and only
if for every there is an such that
Definition:
Let be a metric space. A subset of is said to be closed
if its complement, , is open.
Note that a set can be neither open non closed, like the interval in , and
a set can be both open and closed, something that is true in any metric space for the sets
and .
When thinking about metric spaces, and trying to understand them more intuitively,
the one-dimensional metric space is a little too restrictive to capture the full intuition.
Usually, the picture to have in mind is with its
usual metric, in which is the disk with center and radius , not including the
boundary of the disk, as shown in this illustration:

The discrete metric, which is defined and examined in several exercises in the
reading, is another good example to keep in mind, since it is so different from .
It provides some counterexamples to things that you might have expected to be true in
all metric spaces.
The collection of open sets in a metric space includes and ,
is closed under arbitrary (possibly infinite) union, and is closed under finite intersection.
That is, any union of open sets is open, and any intersection of a finite number of
open sets is open. These properties
mean that the collection of open sets in form a topology for
and make into a topological space. Properties of metric
spaces that can be defined purely in terms of open sets are said to be "topological
properties," because they can be generalized to more general topological spaces.
The collection of closed sets in a metric space includes and ,
is closed under finite union, and is closed under arbitrary intersection.
There is an alternative definition for closed sets, in terms of accumulation points,
which are defined in general metric spaces analogously to their definition in .
Definition:
Let be a metric space, let and let Then is an
accumulation point of if
for every
That is, for any there is at least one element of other than itself, that is within
distance of
Theorem:
Let be a metric space, and let is closed
if and only if every accumulation point of is an element of
For any subset of a metric space , the closure of ,
which we will denote as , is defined to be the set that consists of
together with all accumulation points of . The closure of a set is in fact
a closed set; it is the smallest closed set that contains .
If is already a closed set, then .
If is a metric space and is a subset of , it is natural to make
into a metric space by using the same measure of distance in as is used in
. That is, for , the distance from to in is taken to
be , just like it is in . With this metric, is said to be
a subspace of .
It is important to understand how open and closed sets work in subspaces.
If is a subspace of , a subset of is open in if and
only if there is an open set in such that .
That is, open sets in are intersections of with open setsets of .
Similarly, closed sets in are intersections of with closed subsets of .
For example, the interval can be considered to be a subspace of .
In this subspace, is an open set since it is the intersection of
with (for example) the open subset of . (It can also be seen to
be open by noting that is equal to the open ball of radius about in
If we make into a subspace of , we see that the set
is both open and closed in , since it is the intersection of with both the open
interval and with the closed interval .
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