3.
(a) Closed.
N
c
is open.
(b) Neither open nor closed. Let
r
∈ Q
. Then every neighborhood
N
of
r
contains points
of
Q
and points of
Q
.
(c) Neither open nor closed. Not closed because 0 is an accumulation point of the set which
is not in the set.
Not open because for each
n
every neighborhood of
1
/n
contains
points which are not in the set.
(d)
∪
∞
n
=1
0
,
1
n
=
∅
which is both open and closed.
(e)
{
x
:
x
2
>
0
}
= (
∞
,
0)
∪
(0
,
∞
) – open.
(f)
{
x
:

x

2
≤
3
}
= [

1
,
5] – closed.
4.
(a) If
u /
∈
S
,
then
(
u

, u
)
contains a point of
S
for every
>
0
which implies that
u
∈
S
.
(b) False. 1 = sup [0
,
1] and
1
∈
[0
,
1].
5. Let
x
∈
S
and let
N
(
x,
) be a neighborhood of
x
. Then
N
(
x,
) contains a point
s
1
∈
S
;
the neighborhood of radius
/
2 centered at
x
contains a point
s
2
∈
S
; The neighborhood of
radius
/
3 centered at
x
contains a point
s
3
∈
S
; and so on.
1
Exercises 1.5
1.
(a) True.
(b) False. The closed interval [0
,
1] is compact.
(c) True. Lemma 14.4
(d) False. The set
S
=
{
1
} ∪
(0
,
1)
∪ {
2
}
has a maximum and a minimum but it is not
compact.
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 Fall '08
 Staff
 Topology, Metric space, Sn, Closed set, General topology, ∩TT ∈T