Chapter 3
Topology of the Real Line
“Topology”: the study of qualitative geometric properties: connected, continuous, ...
1
Abstractly, this amounts to studying open sets. In
R
, this is unions of intervals (
a, b
).
In analysis, topology is all about knowing when a sequence converges.
x
n
→
L
⇐⇒
(
U
is an open nbd of
L
=
⇒
x
n
∈
U
for all but finitely many
n
)
.
3.1
Limits and bounds
Not all sequences have a limit, so we need another idea.
Definition 3.1.1.
Let
A
be a nonempty subset of
R
. Then define the
supremum
of
A
to
be the least (smallest) upper bound of
A
. In other words,
a
= sup
A
means:
1.
A
≤
a
, that is,
x
∈
A
=
⇒
x
≤
a
. So
a
is an upper bound of
A
.
2.
A
≤
b
=
⇒
a
≤
b
. So
a
is the smallest upper bound.
If
A
has no upper bound, write sup
A
=
∞
.
Definition 3.1.2.
The infimum of a nonempty set
A
⊆
R
is the greatest lower bound of
A
, defined analogously.
Example 3.1.1.
Let
A
=
{
x
.
.
.
x
2
≤
2
}
and
B
=
{
x
.
.
.
x
2
≥
2
}
. Then
A
is the set of lower
bounds of
B
and
B
is the set of upper bounds of
A
.
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 Fall '11
 Wong
 Topology, Sets, Empty set, Supremum, Order theory, upper bound, α

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