Math 396 Product topology
The aim of this handout is to address two points: metrizability of nite products of metric
spaces, and the abstract characterization of the product topology in terms of universal mapping
properties among topological spaces This latter issue is related to explaining why the denition of
the product topology is not merely ad hocbut in a sense the ight" denition In particular, when
you study topology more systematically and encounter the problem of topologizing innite products
of topological spaces, if you think in terms of the universal property to be discussed below then
you will inexorably be led to the right denition of the product topology for a product of innitely
many topological spaces (it is not what one would naively expect it to be, based on experience with
the case of nite products)
1Metrics on finite products
Let X
1; : : : ; X
dbe metrizable topological spaces The product set
X=X
1
X
d
admits a natural product topology, as discussed in class It is natural to ask if, upon choosing
metrics
j inducing the given topology on each
X
j, we can dene a metric
on Xin terms of
the
j's such that
induces the product topology on X The basic idea is to nd a metric which
describes the idea of \coordinatewise closeness", but several natural candidates leap out, none of
which are evidently better than any others:
max
((x
1; : : : ; x
d)
; (x 0
1 ; : : : ; x 0
d )) = max
1 j d
j(
x
j; x 0
j )
Euc
((x
1; : : : ; x
d)
; (x 0
1 ; : : : ; x 0
d )) = v
u
u
t
d
X
j =1
j(
x
j; x 0
j )
2
1((
x
1; : : : ; x
d)
; (x 0
1 ; : : : ; x 0
d )) = d
X
j =1
j(
x
j; x 0
j )
p((
x
1; : : : ; x
d)
; (x 0
1 ; : : : ; x 0
d )) = 0
@ d
X
j =1
j(
x
j; x 0
j )p 1
A 1
=p
; p 1
When X
j=
R for all j, with
j the usual absolute value metric, these recover the various concrete
norms we've seen on X=Rd Our rst aim will be to show that all of these rather dierentlooking
metrics are at least bounded above and below by a positive multiple of each other (which is the best
we can expect, since they sure aren't literally the same), and so in particular they all dene the
same topology In fact, we will see that the common topology they dene is the product topology We rst axiomatize the preceding examples Let N:R d
! Rbe any norm which satises the
property that on the orthant [0 ;1 )d
with nonnegative coordinates it is a monotonically increasing
function in each individual coordinate when all others are held xed Examples of such N's include
our old friends
k kmax;
k k
Euc;
k k
1;
k k
p(for
p 1)
where we recall that
k(a
1; : : : ; a
n)
k
p = 0
@ d
X
j =1 j
a
jjp 1
A 1
=p
:
1
2
Here is the general theorem which shows that many metrics (including all those mentioned above)
on a product space are bounded above and below by a positive multiple of each other and hence
determine the same theory of open sets, closed sets, and convergence of sequences
Theorem 11 LetN:R d
! Rbe any norm as considered above Then for metric spaces (X
j;
j)
for 1 j d, with product space X=X
1
X
d, the function
N :
X X ! Rdened by
N ((
x
1; : : : ; x
d)
; (x 0
1 ; : : : ; x 0
d )) =
N(
1(
x
1; x 0
1 )
; : : : ;
d(
x
d; x 0
d ))
is a metric on X, and al l such
N 's are bounded above and below by a positive multiple of each
other
Proof Let's rst check that each
N really is a metric Since
Nis