245B, Winter 2009, Assignment 5:
Notes and selected model answers
(Model solutions follow question numbers in
bold
.)
Folland Chapter 4
28.
a.
We must verify that
T
satisfies the axioms of a topology.
1. We always have
π

1
(
∅
) =
∅
and
π

1
(
˜
X
) =
X
, and both of these pre
images are open in
X
, so
∅
,
˜
X
∈ T
;
2. If
U
is any collection of members of
T
, then by definition
{
π

1
(
U
) :
U
∈
U}
is a collection of open subsets of
X
. Therefore
U
∈U
π

1
(
U
)
is a union
of open sets in
X
and so is itself open, but since
π

1
U
=
U
∈U
π

1
(
U
)
this tells us that
π

1
(
U
)
is open, and so
U ∈ T
;
3. Exactly similarly, if
U
is a finite family of members of
T
then
π

1
U
=
U
∈U
π

1
(
U
)
is now a
finite
intersection of open subsets of
X
, so is itself open, and so the
finite intersection
U
also lies in
T
.
b.
Suppose that
Y
is another topological space and
f
:
˜
X
→
Y
a function.
First note that if
U
⊆
˜
X
is open (i.e., it lies in the family
T
that we have shown
above to be a topology) then by the definition of
T
we have that
π

1
(
U
)
is open
1
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in
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Phrased differently, this tells us that
π
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 Spring '10
 tao/analysis
 Topology, Metric space, kN, open subsets, K1 × K2, compact neighbourhood Ki

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