Supplementary Notes for MM509 Topology II
1. The Space of Continuous Functions
Andrew Swann
!
paces of continuous functions f : S
M are important in several applications of
topology. In these notes we will lead up to the situation where S is a compact
topological space and (M; d) is metric space. This is an elaboration of 9.2.10 and
9.2.11 in Sutherland [1].
S
1.1 Bounded functions
Let (M; d) be a metric space and suppose that D is any set. We write
f
B(D; M ) = f : D
! M j f is bounded g
for the space of bounded functions, i.e., functions such that f (D) is a bounded subset
of M .
This carries a metric d1 defined by
d1 (f; g ) = sup d(f (x); g (x)):
x2 D
!
! 1 if and only if fn converges to f
2 N such that for all x 2 D and each
Note that fn
f with respect to d1 as n
uniformly on D, i.e., given " > 0, there is an N
n > N we have d(fn (x); f (x)) < ".
(
)
Proposition 1.1. If (M; d) is complete, then (B(D; M ); d1 ) is complete too.
Proof. Let fn be a Cauchy sequence in (B(D; M ); d1 ). Suppose " > 0. There is an
N N such that
d1 (fn ; fm ) < ";
for all m; n > N .
2
For fixed x
2 M , the sequence
d(fn (x); fm (x))
(fn (x))
is Cauchy in (M; d), since
6 sup d(fn(y); fm(y)) = d1(fn; fm) < ";
y 2D
1.1
for all m; n > N . (1.1)
As (M; d) is complete, the sequence (fn (x)) converges in M and we may define a
function f : D
M by
!
f (x) = lim fn (x):
In particular, for each x
n!1
2 D there is an m x
( )
> N such that
d(fm(x) (x); f (x)) < ":
(1.2)
This implies that for any n > N
d(fn (x); f (x))
6 d(fn(x); fm x (x)) + d(fm x (x); f (x)) < " + " = 2";
( )
( )
by (1.1) and (1.2). As this holds for any x
n > N that
2 D with the same N , we have for each
d1 (fn ; f ) = sup d(fn (x); f (x))
so fn
x2 D
! f with respect to d1 as n ! 1. Moreover,
d(f (x); f (y ))
6 2" < 3";
6 d(f (x); fN (x)) + d(fN (x); fN (y)) + d(fN (y); f (y))
< d(fN (x); fN (y )) + 4";
so fN bounded implies that f is bounded too.
1.2 Continous bounded functions
Let (S; TS ) be a topological space, (M; d) a metric space. We write
Cb
=
f f S ! M j f is continuous and bounded g
:
for the space of continuous functions that are also bounded.
Proposition 1.2. If (M; d) is complete then (Cb ; d1 ) is complete too.
Proof. Let (fn ) be a Cauchy sequence in Cb (S; M ). Then (fn ) is a Cauchy sequence
in B(S; M ) so converges to some bounded function f : S
M . We need to show that
f is continuous.
Fix x S . Given " > 0, we have from fn
f in d1 that there is an N
N such
that
d1 (fn ; f ) < ";
for all n > N :
!
2
!
2
Since fN is continuous at x, there is an open set U in S containing x such that
d(fN (x); fN (y )) < ";
Now we have for each y
d(f (x); f (y ))
2 U that
for all y
2 U:
6 d(f (x); fN (x)) + d(fN (x); fN (y)) + d(fN (y); f (y))
< d1 (f; fN ) + " + d1 (fN ; f ) < 3":
Thus f is continuous at each x
2 S, and hence f 2 C
1.2
b(
S; M ).
If S is compact, then every continuous function f : S
have
! M is bounded.
We thus
Corollary 1.3. For S a compact topological space, and (M; d) a complete metric
space, the space
C(S; M ) = f : S
M f continuous
is complete with respect to d1 .
f
! j
g
References
[1] W. Sutherland, Introduction to metric and topological spaces, Clarendon Press,
Oxford, 1983.
Last revised: 4th February 2008.
1.3