Math 117 – April 26, 2011
9
Topology of the reals
Definition 9.1.
Let
x
∈
R
,
>
0. A neighborhood of
x
of size
is the set
N
(
x,
) =
{
y
∈
R
:

x

y

<
}
.
A deleted neighborhood of
x
of size
is the set
N
*
(
x,
) =
{
y
∈
R
: 0
<

x

y

<
}
.
Clearly
N
*
(
x,
) =
N
(
x,
)
\ {
x
}
. In our case of the set of real numbers
R
,
N
(
x,
) = (
x

, x
+ ),
however the above definition of a neighborhood is more general and works for any metric spaces. Using
the notion of a neighborhood, points in
R
can be classified as interior, boundary or exterior to any
particular subset
S
⊆
R
.
Definition 9.2.
Let
S
⊆
R
. A point
x
∈
R
is called
interior
to
S
, if there is a neighborhood of
x
that
entirely lies in
S
, i.e.
x
∈
N
⊆
S
(
∃
>
0
, N
(
x,
)
⊆
S
). If for every neighborhood of
x N
∩
S
6
=
∅
and
N
∩
(
R
\
S
)
6
=
∅
, then
x
is called a
boundary
point for
S
, and
x
is called
exterior
to
S
, if there exists
a neighborhood of
x
, such that
N
∩
S
=
∅
. The set of interior points is denoted by int
S
, and the set
of boundary points is denoted by bd
S
.
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 Spring '08
 Akhmedov,A
 Topology, Empty set, Metric space, Closed set, General topology

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