18.100B,
Fall
2002,
Solutions
to
Homework
3
Rudin:
(1)
Chapter
2,
Problem
6
Done
in
class
on
Thursday
September
26.
Here
E
⊂
X
is
a
subset
of
a
metric
space
and
E
is
the
set
of
limit
points,
in
X,
of
E.
(a)
Prove
that
E
is
closed.
If
p
∈
X
is
a
limit
point
of
E
then
for
each
r >
0
, B
(
p,
r
)
∩
E
q
is
not
empty.
Since
q
is
a
limit
point
of
E
and
r
−
d
(
p,
q
)
>
0
,
B
(
q,
r
−
d
(
p,
q
))
∩
E
is
an
infinite
set.
By
the
triangle
inequality,
B
(
q,
r
−
d
(
p,
q
))
⊂
B
(
p,
r
) so
B
(
p,
r
)
∩
E
is
also
infinite
and
p
is
therefore
a
limit
point
of
E,
i.e.
p
∈
E
.
Thus
E
contains
each
of
its
limit
points
and
it
is
therefore
closed.
(b)
Prove
that
E
and
E
have
the
same
limit
points.
If
p
is
a
limit
point
of
E
then
it
is
a
limit
point
of
E
since
E
⊂
E.
If
p
1
is
a
limit
point
of
E
then
B
(
p,
n
)
∩
(
E
\{
0
}
)
decreases
with
n
; either it
is
infinite
for
all
n
or
it
is
empty
for
large
n.
We
show
that
the
second
1
case
cannot
occur.
Indeed
this
woould
imply
that
B
(
p,
n
)
∩
(
E
\{
p
}
) is
infinite
for
all
n
and
hence
that
p
is
a
limit
point
of
E
; by the
preceding
result
it
is
then
a
limit
point
of
E
contradicting
the
assumption
that
it
is
not.
Thus
a
limit
point
of
E
is
a
limit
point
of
E.
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 Fall '09
 Frade
 Topology, Metric space, Closure, Closed set, General topology, limit point

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