Math 521  Advanced Calculus I
Instructor: J. Metcalfe
Due: February 17, 2010
Assignment 11
1.
Let
E
0
be the set of all limit points of a set
E
. Prove that
E
0
is closed. Do
E
and
E
0
always
have the same limit points?
For the latter question, the answer is no. Look at
E
=
{
1
/n
:
n
∈
N
}
. Here
E
0
=
{
0
}
, but there are no limit points of
E
0
.
To show that
E
0
is closed, we need to show that it contains its limit points. Thus,
we let
x
be a limit point of
E
0
and we need to show that
x
is a limit point of
E
(i.e. is
in
E
0
). For a given
ε >
0, we need to show that there is a point
y
∈
E
with
y
6
=
x
so
that
y
∈
V
ε
(
x
). For this same
ε
, since
x
is a limit point of
E
0
, we know that there is an
element
z
∈
E
0
with
z
6
=
x
so that
z
∈
V
ε/
2
(
x
). I.e.

z

x

< ε/
2. Since
z
∈
E
0
(i.e.
z
is
a limit point of
E
), there is an element
y
∈
E
with
y
6
=
z
so that
y
∈
V

x

z

/
2
(
z
). Then,

x

y
 ≤ 
x

z

+

z

y

<
(
ε/
2) +

x

z

/
2
<
3
ε/
4
< ε.
Thus,
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 Spring '10
 MetCalfe
 Calculus, Closure, General topology, interior point, J. Metcalfe, Vε

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