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Problem 92.
{
K
c
n
:
n
≥
1
}
and 0 form an open covering of
K
1
. Suppose
∪
N
1
K
c
n
i
∪
O
⊇
K
1
. Then
K
c
n
N
∪
O
⊇
K
1
⊇
K
n
N
. Hence
O
⊇
K
n
N
.
Problem 94.
Let
A,B
be closed subsets. Note that
A,B
are compact. Fix
x
∈
A
. By
Problem 9.3 there are disjoint open sets
O
x
and
U
x
with
x
∈
O
x
and
B
⊆
U
x
.
Since
A
is compact, we can ﬁnd
{
O
x
i
: 1
≤
i
≤
N
}
which covers
A
.
Let
O
A
=
S
n
1
O
x
i
and
O
B
=
T
n
1
U
x
i
.
Problem 916.
Urysohn’s Lemma and Probem 9.4 imply
g
is onetoone. Now use
Proposition 9.5 and Tychonoﬀ Theorem.
Problem 919 a.
By Problem 9.18 there is an open set
O
⊇
K
with
¯
O
compact. By
Problem 9.4 and Urysohn’s Lemma there is a continuous function
f
on
¯
O
with
f

K
= 1 and
f

¯
O
\
O
= 0. Extend
f
by setting
f
= 0 on
¯
O
c
. Hence
f

O
c
= 0. Now use Proposition 8.3
with
A
=
O
c
and
B
=
¯
O
to show
f
is continuous on
X
. Note that
{
x
:
f
(
x
)
6
= 0
} ⊆
O
⊆
¯
O
is thus compact.
b.
Suppose
K
=
T
∞
1
O
i
. By Problem 9.18 we may assume each
¯
O
i
is compact.
Otherwise consider
T
∞
1
(
O
i
∩
O
), where
O
is given in Problem 9.18. By Part a there
are continuous functions
f
i
with
0
≤
f
i
≤
1
, f
i

K
= 1 and
f
i

O
c
i
= 0
.
Let
g
=
∑
∞
1
f
i
/
2
i
. Then
g
is continuous as the series converges uniformly. If
x /
∈
K
then
x
∈
O
c
i
for some
i
. Hence
f
i
(
x
) = 0 and then
g
(
x
)
<
1.
Problem 926.
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This note was uploaded on 03/23/2010 for the course MATH 515 taught by Professor Staff during the Spring '08 term at Iowa State.
 Spring '08
 Staff
 Sets

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