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Corollary 5.5.10 is useful because it says that when spaces are compact then it is
easier to show that maps are homeomorphisms. Normally, one would have to show
that their inverse is continuous, but we do not need to prove that here.
5.5.11. Corollary.
Let
X
be a nonempty compact space and assume that f :
X
Æ
R
is a continuous map. Then f attains both its maximum and minimum value on
X
, that
is, there are
x
1
and
x
2
in
X
so that
f
()
£
x
f
()
£
x
f
()
x
1
2
for all
x
X
.
Proof.
By Theorem 5.5.8, f(
X
) is a compact subset of
R
and hence is closed and
bounded by Theorem 5.5.6. Therefore, sup f(
X
) and inf f(
X
) belong to f(
X
), proving
the existence of
x
1
and
x
2
.
As another example of why compactness is nice, recall Example 4.2.7, which
showed that a continuous function need not be uniformly continuous. The next
theorem states that if the domain of the function is compact then this does not
happen. The proof of the theorem uses the lemma below that we state separately
because it has other applications.
5.5.12. Lemma.
(Lebesgue Covering Lemma) Let (
X
,d) be a compact metric space.
Given any open cover W of
X
, there is a d>0, so that if
p
,
q
X
and d(
p
,
q
) <d, then
there is a set
U
in W that contains both
p
and
q
. (The number d is called a
Lebesgue
number
for W.)
Proof
.
See [Eise74].
5.5.13. Theorem.
Let f :
X
Æ
Y
be a continuous map between metric spaces. If
X
is
compact, then f is uniformly continuous.
Proof
.
See [Eise74].
We finish this section with the definition for one well-known construction for
“compactifying” certain noncompact spaces.
Definition.
A Hausdorff space
X
is said to be
locally compact
if every point has a
compact neighborhood.
Clearly, every compact space is locally compact.
R
n
is locally compact but not
compact.
5.5.14. Theorem.
Let
X
be a noncompact, locally compact topological space with
topology T. Let
•
=»
{}
,
XX
