1
2
1
𝑆
𝐵
1
/
2
((2
,
0))
Figure 1.2: Open ball about (2
,
0) relative to
𝑋
1.95
Assume
𝑆
is connected.
Suppose
𝑆
is not an interval.
This implies that there
exists numbers
𝑥, 𝑦, 𝑧
such that
𝑥 < 𝑦 < 𝑧
and
𝑥, 𝑧
∈
𝑆
while
𝑦 /
∈
𝑆
. Then
𝑆
= (
𝑆
∩
(
−∞
, 𝑦
))
∪
(
𝑆
∩
(
𝑦,
∞
))
represents
𝑆
as the union of two disjoint open sets (relative to
𝑆
), contradicting the
assumption that
𝑆
is connected.
Conversely, assume that
𝑆
is an interval. Suppose that
𝑆
is not connected. That is,
𝑆
=
𝐴
∪
𝐵
where
𝐴
and
𝐵
are nonempty disjoint closed sets. Choose
𝑥
∈
𝐴
and
𝑧
∈
𝐵
.
Since
𝐴
and
𝐵
are disjoint,
𝑥
∕
=
𝑧
. Without loss of generality, we may assume
𝑥 < 𝑧
.
Since
𝑆
is an interval, [
𝑥, 𝑧
]
⊆
𝑆
=
𝐴
∪
𝐵
. Let
𝑦
= sup
{
[
𝑥, 𝑧
]
∩
𝑆
}
Clearly
𝑥
≤
𝑦
≤
𝑧
so that
𝑦
∈
𝑆
. Now
𝑦
belongs to either
𝐴
or
𝐵
. Since
𝐴
is closed in
𝑆
,
[
𝑥, 𝑧
]
∩
𝐴
is closed and
𝑦
= sup
{
[
𝑥, 𝑧
]
∩
𝑆
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 Fall '10
 Dr.DuMond
 Macroeconomics, Topology, Metric space, Topological space, Michael Carter, Foundations of Mathematical Economics

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