Economics 204
Lecture 4–Thursday, July 31, 2008
Section 2.4, Open and Closed Sets
Defnition 1
Let (
X, d
) be a metric space. A set
A
⊆
X
is
open
if
∀
x
∈
A
∃
ε>
0
B
ε
(
x
)
⊆
A
Aset
C
⊆
X
is
closed
if
X
\
C
is open.
Example:
(
a, b
)i
sopenintheme
t
r
icspace
E
1
(
R
with the usual Euclidean metric). Given
x
∈
(
a, b
),
a<x<b
.L
e
t
ε
=m
in
{
x
−
a, b
−
x
}
>
0
Then
y
∈
B
ε
(
x
)
⇒
y
∈
(
x
−
ε, x
+
ε
)
⊆
(
x
−
(
x
−
a
)
,x
+(
b
−
x
))
=(
a, b
)
so
B
ε
(
x
)
⊆
(
a, b
), so (
a, b
)isopen
.
Notice that
ε
depends on
x
;inpar
t
icu
lar
,
ε
gets smaller as
x
nears the boundary of the set.
Example:
In
E
1
,[
a, b
]isc
losed
.
R
\
[
a, b
]=(
−∞
,a
)
∪
(
b,
∞
) is a union of two open sets, which must be
open .
.. .
Example:
In the metric space [0
,
1], [0
,
1] is open. With [0
,
1] as the underlying metric space,
B
ε
(0) =
{
x
∈
[0
,
1] :

x
−
0

<ε
=[0
,ε
).
Thus, openness and closedness depend on the underyling metric space as well
as on the set.
Example:
Most sets are neither open nor closed. For example, in
E
1
,[0
,
1]
∪
(2
,
3) is neither open nor
closed.
Example:
An open set may consist of a single point. For example,
1