11. Let
S
=
Q
. Then
int S
=
∅
, and
bd S
=
R
.
12. Think about a cell in the space
R
3
, its membrane is the “boundary” of the cell. For a
region in
R
2
or a solid region in
R
3
, we can observe its boundary. The above definition
can be used for these cases. In general, the boundary of a region could be very very
complicated, but it all can be defined by the above definition.
Open sets and closed sets
Let
S
⊆
R
.
Definition
1.
S
is
open
if every point
x
∈
S
is an interior point of
S
.
2.
S
is
closed
if
bd S
⊆
S
.
[Examples and remarks]
1. The basic model for “open sets” is an open interval (
a, b
). This may be why we use
the name “open.” The basic model for “closed set” is a closed interval [
a, b
]. This may
be why we use the name “closed.”
2. The most significance of the notion of “open sets” is that it can be used to define
“continuous functions” between general sets. For example, for a function
f
:
R
→
R
,
we can define that
f
is continuous
⇐⇒
f
-
1
(
U
) is an open set,
for any open set
U
.
(1)
where
f
-
1
(
U
) =
{
x
∈
R
|
f
(
x
)
∈
U
}
.
3. For any finite numbers
a
and
b
, (
a, b
] and [
a, b
) are neither open nor closed.
4. Let
S
=
{-
1
} ∪
(0
,
2) is neither open nor closed.
5. The set of a finite number of points
S
=
{
a
1
, a
2
, ..., a
n
}
is closed.
6. (
a,
∞
) is open, and [
a,
∞
) is closed.
7. (
-∞
, b
) is open, and (
-∞
, b
] is closed.
8. Let
S
=
{
1
,
1
2
,
1
3
, ...,
1
n
, ...
}
. Then
S
is either open, nor closed.
9. Let
S
=
N
. Then
S
is not open but is closed.
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