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Homework 4 for
MATH 104
Solutions
Problem 1
Pugh, p.116, #23
Solution.
Assume
h
:
Q
→
N
is a homeomorphism. Note that
N
is homeomorphic to a discrete space: Consider
(
N
,
d
) where
d
is the discrete metric and let
f
=
id.
(1
/
n
) is a convergent sequence in
Q
. Since
h
is a bijection, all terms of
h
(1
/
n
) are distinct. In particular, the
sequence
h
(1
/
n
) is not eventually constant and thus, by 2.12 (c), does not converge in
N
. Hence
h
is not continuous,
contradiction.
±
Problem 2
Pugh, p.116, #18
Solution.
(a) Let
f
:
M
→
N
be an isometry. We show that
f
has the
ε

δ
property. If
ε >
0, let
δ
=
ε
. Then for all
p
∈
M
,
d
(
p
,
q
)
< δ
implies
d
(
f
(
p
)
,
f
(
q
))
=
d
(
p
,
q
)
< δ
=
ε.
(b) Since any isometry is by deﬁnition a bijection, it su
ﬃ
ces to show that
f

1
:
N
→
M
is continuous. Let
ε >
0. Again, put
δ
=
ε
. Let
z
∈
N
, and assume
y
∈
N
is such that
d
(
z
,
y
)
< δ
. Let
p
,
q
be such that
f
(
p
)
=
z
,
f
(
q
)
=
y
Then
d
(
f

1
(
z
)
,
f

1
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 Fall '08
 RIEMAN
 Math

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