Solutions for HW 3
Problem 47: 4. Sets consisting of one point are obviously connected. Let
E
⊂
R
\
Q
contain at least two points
a < b
. Then there exist a rational number
r
with
a < r < b
. Let
G
1
= (
∞
,r
) and
G
2
= (
r,
∞
). Then
G
1
,G
2
are disjoint open
sets with union
R
\ {
r
}
. Now
a
∈
E
∩
G
1
and
b
∈
E
∩
G
2
. Hence
E
∩
G
1
6
=
∅
and
E
∩
G
2
6
=
∅
, but
E
∩
G
1
∩
G
2
=
∅
and
G
1
∪
G
2
⊃
E
. Hence
E
is not connected.
Problem 47: 5. Let
G
1
and
G
2
be open sets such that
G
1
∩
E
6
=
∅
,
G
2
∩
E
6
=
∅
,
G
1
∩
G
2
∩
E
=
∅
, and
G
1
∪
G
2
⊃
E
. Let
x
∈
G
1
∩
E
. Then there exists a
r >
0
such that
B
(
x
;
r
)
⊂
G
1
. Also
B
(
x
;
r
)
\ {
x
} ∩
E
6
=
∅
. Hence
G
1
∩
E
6
=
∅
. Similarly
G
1
∩
E
6
=
∅
. As
E
∩
G
1
∩
G
2
=
∅
and
G
1
∪
G
2
⊃
E
are obvious, it follows that
E
is not connected. Contradiction. If
E
= (0
,
1)
∩
(1
,
2), then
E
= [0
,
2] is connected,
but
E
is not.
Problem 47: 6. Let
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This note was uploaded on 02/05/2012 for the course MATH 703 taught by Professor Schep during the Fall '11 term at South Carolina.
 Fall '11
 Schep
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