Proposition 5.3.3: Connected Sets in R are Intervals
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If
S is any connected subset of
R then
S must be some interval.
Context
Proof:
If
S is not an interval, then there exists
a, b S and a
point
t between
a and
b such that
t is
not in
S. Then define the two sets

U = (  , t ) and
V = ( t, )
Then
U S #
0
(because it contains
{ a }) and
V S #
0
(because it contains
{ b }), and clearly
(
U S)
(
V S) =
0.
Finally, because
t is not contained in
S, we know
that
(
U S)
(
V S) =
S.
Hence, we have found the required sets
U and
V to
disconnect
S. So, we have proved that if a set is not an
interval it is disconnected. That is equivalent to saying that if it
is connected, it must be an interval.