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Midterm 2 Solutions for MATH 104, Section 1
November 5, 2009
Show your work and justify your answers. 48 points total
1. Deﬁne the following:
(a) (4 pts.) a
metric space
;
(b) (4 pts.) a
disconnected
metric space.
2. (4 pts.) State the inheritance principle.
(see the book for answers to these problems)
3. (12 pts.) Consider a function
f
:
M
→
N
, where
M
and
N
are metric spaces. Prove
that if
f
is continuous then its graph is closed as a subset of
M
×
N
. Recall that the
graph is
graph
(
f
) =
{
(
p,y
)
∈
M
×
N
:
y
=
f
(
p
)
}
.
Solution:
Let
F
:
M
×
N
→
R
be deﬁned by
F
(
p,y
) =
d
(
f
(
p
)
,y
). This is a
composition of continuous functions (the distance function
d
is continuous by Ex
ercise 2.40), and therefore is continuous by Corollary 2.3. Also, if
F
(
p,y
) = 0, then
d
(
f
(
p
)
,y
) = 0 which means that
f
(
p
) =
y
by the positive deﬁniteness of the metric
d
.
Then
graph
(
f
) =
{
(
p,y
)
∈
M
×
N
:
F
(
p,y
) = 0
}
=
F

1
[
{
0
}
]. Since
{
0
}
is a closed
subset of
R
,
F

1
[
{
0
}
] is a closed subset of
M
×
N
by Theorem 10(ii), so
graph
(
f
) is
closed.
4. Either prove that the statement is true, or prove it is false.
(a) (6 pts.) There exists a continuous surjective map from the circle
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 Fall '08
 RIEMAN
 Math

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