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Econ 204 Section 5
GSI: Hui Zheng
Key Words
Open Set, Closed Set, Interior, Closure, Exterior, Boundary, Limits of Function, Continuity,
Uniform Continuity, Lipschitz Functioins, Homemorphism
Section 5.1
Open Set and Closed Set
(
X;d
)
be a metric space. A set
A
±
X
is
open
if
8
x
2
A
9
" >
0
B
"
(
x
)
±
A
. A set
C
±
X
is
closed
if
X
n
C
is open.
int
A
: the
interior
of
A
, the largest open set contained in
A
(the union of all open sets
contained in
A
)
A
: the
closure
of
A
, the smallest closed set containing
A
(the intersection of all closed
sets containing
A
)
ext
A
: the
exterior
of
A
, the largest open set contained in
X
n
A
@A
: the
boundary
of
A
,
(
X
n
A
)
T
A
Lecture 4 Theorem 2: Let
(
X;d
)
collection of open sets is open.
Lecture 4 Theorem 4: A set
A
in a metric space
(
X;d
)
is closed if and only if
f
x
n
g ²
A
,
f
x
n
g !
x
2
X
)
x
2
A
Openness and closedness depend on the underlying metric space as well as on the set.
Theorem 2 is useful for proving that a set is open.
Theorem 4 is useful for proving that a set is not closed. Unfortunately, using this theorem
to prove that a set is closed is considerably more di¢ cult, but still possible in many cases.
Example 5.1.1
Prove the statement or give a counterexample.
Solution:
False. Let
A
n
= (
³
1
;
1
n
)
.
\
1
n
=1
A
n
= (
³
1
;
0]
which is neither open nor closed. Notice
that we can express a closed interval in
R
as the intersection of open intervals.
[
a;b
] =
\
1
n
=1
(
a
³
1
n
; b
+
1
n
)
Example 5.1.2
Prove that the intersection of an arbitrary collection of closed and that the
Solution: From de Morgan±s law we know that
:
(
A
V
B
) =
:
A
W
:
B
. Then the result
Example 5.1.3
State whether the following sets are open, closed, both, or neither:
1.
f
(
x;
0)
j
x
2
(0
;
1)
g
in
R
2
2.
f
(
x;y;z
)
j
0
´
x
+
y
´
1
;z
= 0
g
in
R
3
3.
f
1
n
j
n
2
N
g
in
R
4.
f
1
n
j
n
2
N
g
in
(0
;
1
)
5.
Q
in
R
Solution:
1. Neither 2. Closed 3. Neither 4. Closed 5. Neither
Example 5.1.4
What is the closure of
Q
(the set of all rational numbers)
Solution:
Q
=
R
.
1
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View Full DocumentIf
8
a;b
2
R
a < b
)
Q
\
[
a;b
] = [
a;b
]
, then
Q
\
R
=
[
n
2
Z
f
Q
\
[
n; n
+1] =
[
n
2
Z
[
n:n
+1] =
R
:
So our job is to show
Q
\
[
a;b
] = [
a;b
]
. Consider the open interval
(
a;b
)
Q
is closed, hence
Q
c
is open. So
(
a;b
)
\
Q
c
is open. If
(
a;b
)
\
Q
c
6
=
;
, then there exists
x
2
(
a;b
)
\
Q
c
and
" >
0
, such that
(
x
"; x
+
"
)
±
(
a;b
)
\
Q
c
. By the density of the
rationals in
R
, there exists
q
2
Q
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 Summer '08
 ANDERSON

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