Economics 204
Lecture 4–August 3, 2006
Revised 8/4/06, Revisions indicated by **
Section 2.4, Open and Closed Sets
Definition 1
Let (
X, d
) be a metric space. A set
A
⊆
X
is
open
if
∀
x
∈
A
∃
ε>
0
B
ε
(
x
)
⊆
A
A set
C
⊆
X
is
closed
if
X
\
C
is open.
Example:
(
a, b
) is open in the metric space
E
1
(
R
with the usual
Euclidean metric). Given
x
∈
(
a, b
),
a < x < b
. Let
ε
= min
{
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
) is open.
Notice that
ε
depends on
x
; in particular,
ε
gets smaller as
x
nears the boundary of the set.
Example:
In
E
1
, [
a, b
] is closed.
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:
In any metric space (
X, d
) both
∅
and
X
are open, and
both
∅
and
X
are closed.
To see that
∅
is open, note that the
1