Graphics Reference
In-Depth Information
to answer such a question yet and will have to wait until Chapter 7, but the reader
may appreciate one glimpse into the future. Given a map
1
1
f:
SS
Æ
,
the
degree of f
is, intuitively, the number of times that f winds the circle around itself.
(In Section 7.5.1 we shall give another definition of the degree of f.) Define
1
1
f
n
:
SS
Æ
by f
n
(cos q,sin q) = (cos nq,sin nq). Then f
n
has degree n. It turns out that all maps of
the circle to itself are homotopic to one of these maps and two maps are homotopic
only if they have the same degree, so that there is a bijection between the homotopy
classes of maps of the circle to itself and the integers.
5.8
Constructing Continuous Functions
There are many situations where one wants to define continuous functions on a
topological space satisfying certain properties. This brief section describes two very
fundamental theorems that deal with the existence of certain functions, which in turn
can be used to construct many other functions. We shall give one application having
to do with the existence of partitions of unity at the end of the section.
For a topological space to have the continuous functions we want it needs
to satisfy a special property. It is worth isolating this property and giving it a
name.
Definition.
A topological space
X
is said to be
normal
if, given two disjoint
closed sets
A
and
B
in
X
, there exist disjoint open sets containing
A
and
B
,
respectively.
The condition that a space be normal is somewhat technical, like being Hausdorff,
but fortunately the spaces of interest to us satisfy this property.
5.8.1. Theorem.
(1) Any metrizable space is normal.
(2) Any compact Hausdorff space is normal.
Proof.
See [Eise74].
5.8.2. Theorem.
(The Urysohn Lemma) Let
X
be a normal space and assume that
A
and
B
are two closed subsets of
X
. Then there exists a continuous function f :
X
Æ
[0,1] such that f takes the value 0 on
A
and 1 on
B
.
Proof.
See [Jäni84].
