Graphics Reference
In-Depth Information
{
(
)
}
nn n
-+
11
,
Œ
Z
of
R
.
It would be good to know what condition on a topological space guarantees the
existence of partitions of unity.
Definition.
A cover of a space is
locally finite
if every point in the space has a neigh-
borhood that meets only finitely many elements of the cover. A Hausdorff space is said
to be
paracompact
if every open cover admits a locally finite subcover.
5.8.5. Lemma.
Every paracompact space is normal.
Proof.
See [Jäni84].
5.8.6. Theorem.
A Hausdorff space is paracompact if and only if every open cover
admits a partition of unity subordinate to it.
Proof.
The only hard part is showing that paracompact implies the existence of the
stated partitions of unity. Because of Lemma 5.8.5 one can use Urysohn's lemma to
construct the desired partition of unity. See [Jäni84].
The next obvious question is: which spaces are paracompact?
5.8.7. Theorem.
The following types of topological spaces are paracompact:
(1) Compact Hausdorff spaces
(2) Topological manifolds
(3) Metrizable spaces
Proof.
Part (1) is trivial. Part (2) is also not hard. For (3) see [Schu68].
The Topology of P
n
5.9
Projective space
P
n
is not only one of the really important spaces in mathematics but
it also serves as an excellent example of a nontrivial topological space. This section
looks at its purely topological properties. We shall return to it later in Chapter 8 to
look at its manifold properties and again in Chapter 10 where its algebraic properties
come to the fore.
Recall the (set theoretic) definition of
P
n
given in Section 3.4, namely,
(
)
n
n
+1
(5.11)
where ~ is the equivalence relation on
R
n+1
-
0
defined by
p
~ c
p
, for c π 0. In Chapter
3 we did not say anything about its topology, but actually, when we talk about
P
n
as
a topological space, we always assume that it has been given the quotient space top-
ology that is defined by equation (5.11). What does this space really “look” like top-
PR 0
=
-
~,
