4. Is the set of all irrational real numbers countable?
Solution:
If
R

Q
is countable, then
R
1
= (
R

Q
)
Q
is countable,
a contradiction. Thus
R

Q
is uncountable.
5. Construct a bounded set of real numbers with exactly three limit points.
Solution:
Put
A
=
{
1
/n
:
n
∈
N
}
{
1 + 1
/n
:
n
∈
N
}
{
2 + 1
/n
:
n
∈
N
}
.
A
is bounded by 3, and
A
contains three limit points  0, 1, 2.
6. Let
E
be the set of all limit points of a set
E
.
Prove that
S
is
closed. Prove that
E
and
E
have the same limit points. (Recall that
E
=
E
E
.) Do
E
and
E
always have the same limit points?
Proof:
For any point
p
of
X

E
, that is,
p
is not a limit point
E
,
there exists a neighborhood of
p
such that
q
is not in
E
with
q
=
p
for
every
q
in that neighborhood.
Hence,
p
is an interior point of
X

E
, that is,
X

E
is open, that
is,
E
is closed.
Next, if
p
is a limit point of
E
, then
p
is also a limit point of
E
since
E
=
E
E
. If
p
is a limit point of
E
, then every neighborhood
N
r
(
p
)
of
p
contains a point
q
=
p
such that
q
∈
E
. If
q
∈
E
, we completed
the proof. So we suppose that
q
∈
E

E
=
E

E
. Then
q
is a limit
point of
E
. Hence,
N
r
(
q
)
where
r
=
1
2
min(
r

d
(
p, q
)
, d
(
p, q
)) is a neighborhood of
q
and contains
a point
x
=
q
such that
x
∈
E
. Note that
N
r
(
q
) contains in
N
r
(
p
)
{
p
}
.
That is,
x
=
p
and
x
is in
N
r
(
p
). Hence,
q
also a limit point of
E
. Hence,
E
and
E
have the same limit points.
2