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Answers, Assignment #10
Exercises 1.4
1. (a) True. If
s
∈
int
S
, then
s
does not belong to bd
S
.
(b) False. Let
S
=(0
,
1). Then bd
S
=
{
0
,
1
}
which is not a subset of
S
.
(c) False.
S
=[0
,
1], bd
S
=
{
0
,
1
}
,
S
±
=bd
S
.
(d) True. Suppose
s
∈
S
.I
f
s/
∈
int
S
, then every neighborhood of
s
must intersect
S
c
which implies
s
∈
bd
S
.
(e) True.
s
∈
S
implies
N
∩
S
±
=
∅
and
N
∩
S
c
±
=
∅
for every neighborhood
N
of
s
.
t
∈
S
c
implies
N
∩
S
c
±
=
∅
and
N
∩
(
S
c
)
c
=
N
∩
S
±
=
∅
for every neighborhood
N
of
t
. Therefore bd
S
S
c
.
(f) False.
S
,
1], bd
S
=
{
0
,
1
}⊂
S
.
2. (a) True. Let
z
∈
N
(
x, ±
). Let
δ
= min
{
z

x

,

z

(
x
+
±
)

,

z

(
x

±
)
}
. Then
N
(
z,δ
)
⊂
N
(
x, ±
).
(b) True. Theorem 10 (a).
(c) False.
±
∞
n
=1
²
1
n
,
1

1
n
³
,
1).
(d) False.
´
∞
n
=1
(

1
n
,
1+
1
n
)
,
1].
(e) True. Let
T
be a collection of closed sets. Then [
∩
T
T
∈T
]
c
=
∪
T
c
t
∈T
.
T
closed implies
T
c
is open and the union of a collection of open sets is open. Therefore, [
∩
T
T
∈T
]
c
is
open and
∩
T
T
∈T
is closed.
(f) False. The set of real numbers is both open and closed.
3. (a) Closed.
N
c
is open.
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 Spring '08
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