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1st December 2004
Munkres
§
17
Ex. 17.3.
A
×
B
is closed because its complement
(
X
×
Y
)

(
A
×
B
) = (
X

A
)
×
Y
∪
X
×
(
Y

B
)
is open in the product topology.
Ex. 17.6.
(a)
.
If
A
⊂
B
, then all limit points of
A
are also limit points of
B
, so [Thm 17.6]
A
⊂
B
.
(b)
.
Since
A
∪
B
⊂
A
∪
B
and
A
∪
B
is closed [Thm 17.1], we have
A
∪
B
⊂
A
∪
B
by (a).
Conversely, since
A
⊂
A
∪
B
⊂
A
∪
B
, we have
A
⊂
A
∪
B
by (a) again. Similarly,
B
⊂
A
∪
B
.
Therefore
A
∪
B
⊂
A
∪
B
. This shows that closure commutes with
ﬁnite
unions.
(c)
.
Since
S
A
α
⊃
A
α
we have
S
A
α
⊃
A
α
by (a) for all
α
and therefore
S
A
α
⊃
S
A
α
. In general
we do not have equality as the example
A
q
=
{
q
}
,
q
∈
Q
, in
R
shows.
Ex. 17.8.
(a)
.
By [Ex 17.6.(a)],
A
∩
B
⊂
A
and
A
∩
B
⊂
B
, so
A
∩
B
⊂
A
∩
B
. It is
not
true in general
that
A
∩
B
=
A
∩
B
as the example
A
= [0
,
1),
B
= [1
,
2] in
R
shows. (However, if
A
is open and
D
is dense then
A
∩
D
=
A
).
(b)
.
Since
T
A
α
⊂
A
α
we have
T
A
α
⊂
A
α
for all
α
and therefore
T
A
α
⊂
T
A
α
. (In fact, (a) is
a special case of (b)).
(c)
.
Let
x
∈
A

B
. For any neighborhood of
x
,
U

B
is also a neighborhood of
x
so
U
∩
(
A

B
) = (
U

B
)
∩
A
⊃
(
U

B
)
∩
A
6
=
∅
since
x
is in the closure of
A
[Thm 17.5]. So
x
∈
A

B
. This shows that
A

B
⊂
A

B
. Equality
does not hold in general as
R

{
0
}
=
R
 {
0
}
$
R
 {
0
}
=
R
.
Just to recap we have
(1)
A
⊂
B
⇒
A
⊂
B
(
A
⊂
B
,
B
closed
⇒
A
⊂
B
)
(2)
A
∪
B
=
A
∪
B
(3)
A
∩
B
⊂
A
∩
B
(
A
∩
D
=
A
if
D
is dense.)
(4)
S
A
α
⊃
S
A
α
(5)
T
A
α
⊂
T
A
α
(6)
A

B
⊂
A

B
Dually,
(1)
A
⊂
B
⇒
Int
A
⊂
Int
B
(
A
⊂
B
,
A
open
⇒
A
⊂
Int
B
)
(2) Int (
A
∩
B
) = Int
A
∩
Int
B
(3) Int (
A
∪
B
)
⊃
Int
A
∪
Int
B
These formulas are really the same because
X

A
=
X

Int
A,
Int (
X

A
) =
X

A
Ex. 17.9.
[Thm 19.5] Since
A
×
B
is closed [Ex 17.3] and contains
A
×
B
, it also contains the
closure of
A
×
B
[Ex 17.6.(a)], i.e.
A
×
B
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This note was uploaded on 01/12/2011 for the course MATH 110 taught by Professor Brown during the Fall '08 term at Arizona Western College.
 Fall '08
 Brown
 Topology

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