Economics 204
Lecture 7–Tuesday, August 4, 2009
Note
: In this set of lecture notes,
¯
A
refers to the closure of
A
.
Section 2.9, Connected Sets
Definition 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.
Example:
[0
,
1) and [1
,
2] are disjoint but not separated:
[0
,
1)
∩
[1
,
2] = [0
,
1]
∩
[1
,
2] =
{
1
}
=
∅
[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)
A set
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
.
Proof:
First, we show that
S
connected implies that
S
is an interval. We do this by proving the contra-
positive: if
S
is not an interval, it is not connected. If
S
is not an interval, find
x, y
∈
S, x < z < y, z
∈
S
1