Graphics Reference
In-Depth Information
Figure 5.9.
Attaching a handle to a disk.
B
B
D»
f
H
H
f
f
H
D
D
f(B)
f(B)
Definition.
Let
X
be a set and let
A
i
be subsets of
X
that already possess a topology
(we allow either a finite or infinite set of indices i). Assume
(1) the topologies of
A
i
and
A
j
agree on
A
i
«
A
j
, and either
(2)
A
i
«
A
j
is always open in both
A
i
and
A
j
,
or
(2¢)
A
i
«
A
j
is always closed in
A
i
and
A
j
.
The
weak topology
T on
X
determined by the topologies of the spaces
A
i
is defined by
{
}
T
=Õ
UXUA
«
is open in
A
for all i
.
i
i
5.4.8. Theorem.
Using the notation in the definition of the weak topology, the fol-
lowing holds:
(1) The weak topology is a topology for
X
.
(2) A subset
A
of
X
is closed in the weak topology if and only if
A
«
A
i
is closed
for all i.
(3) The subsets
A
i
will themselves be open subsets of
X
in the weak topology if
condition (2) in the definition held and closed subsets if (2¢) held.
Proof.
Easy.
Next, one often wants to take the product of topological spaces. Since we want to
end up with a topological space, we need to define a product topology. We shall build
on what we know for metric spaces.
5.4.9. Theorem.
If
X
i
, 1 £ i £ k, are topological spaces, then the collection of subsets
{
}
UU
¥¥ ¥
...
UU
is open in
X
i
2
k
i
i
form the base of a unique topology on
X
1
¥
X
2
¥ ...¥
X
k
.
Proof.
One simply has to show that these subsets satisfy conditions (1) and (2) in
Theorem 5.3.4. See [Eise74].
Definition.
If
X
i
, 1 £ i £ k are topological spaces, then the topology on the product
set
X
=
X
1
¥
X
2
¥ ...¥
X
k
described in Theorem 5.4.9 is called the
product topology
on
X
.
