Economics 204
Lecture 7–Tuesday, August 4, 2009
Revised 8/5/09, Revisions indicated by ** and
Sticky Notes
Note
: In this set of lecture notes,
¯
A
refers to the closure of
A
.
Section 2.9, Connected Sets
Defnition 1
Two sets
A, B
in a metric space are
separated
if
¯
A
∩
B
=
A
∩
¯
B
=
∅
A set in a metric space is
connected
if it cannot be written as the
union of two nonempty separated sets.
**
Remark:
In other texts, you will see the following equivalent
deFnition: A set
Y
in a metric space
X
is connected if there do
not exist open sets
A
and
B
such that
A
∩
B
=
∅
,
Y
⊆
A
∪
B
and
A
∩
Y
6
=
∅
and
B
∩
Y
6
=
∅
.
Example:
[0
,
1) and [1
,
2] are disjoint but not separated:
[0
,
1)
∩
[1
,
2] = [0
,
1]
∩
[1
,
2] =
{
1
} 6
=
∅
[0
,
1) and (1
,
2] are separated:
[0
,
1)
∩
(1
,
2] = [0
,
1]
∩
(1
,
2] =
∅
[0
,
1)
∩
(1
,
2] = [0
,
1)
∩
[1
,
2] =
∅
Note that
d
([0
,
1)
,
(1
,
2]) = 0 even though the sets are separated.
Note that separation does
not
require that
¯
A
∩
¯
B
=
∅
.
[0
,
1)
∪
(1
,
2]
is not connected.
Theorem 2 (9.2)
Aset
S
of real numbers is connected if and
only if it is an interval, i.e. given
x, y
∈
S
and
z
∈
(
x, y
)
,then
z
∈
S
.
1