147 HW3 Solutions
17.6) Let
A
,
B
and
A
α
denote subsets of a space
X
. Prove the following:
(a) If
A
⊂
B
, then
A
⊂
B
.
Solution: Let
C
be some closed subset of
X
. If
B
⊂
C
then
A
⊂
C
so by the deﬁnition of
U
as the
intersection of all closed subsets of
X
containing
U
, we have that
A
⊂
B
.
(b)
A
∪
B
=
A
∪
B
.
Solution:
A
∪
B
⊂
C
if and only if
A
⊂
C
and
B
⊂
C
. Using part (a) we have
A
⊂
A
∪
B
⊃
B
,
and thus
A
∪
B
⊃
A
∪
B
. But
A
∪
B
⊂
A
∪
B
, and the latter set is closed (ﬁnite unions
of closed sets are closed), so
A
∪
B
⊂
A
∪
B
by the deﬁnition of
A
∪
B
.
(c)
S
A
α
⊃
S
A
α
; give an example of where equality fails.
Solution: If
S
A
α
⊂
C
, then for any
α
,
A
α
⊂
C
. Thus
S
A
α
⊃
A
α
for every
α
, giving us that
S
A
α
⊃
S
A
α
. Note: only
ﬁnite
unions of closed sets are guaranteed to be closed.
For an example of equality failing, consider
A
n
= (
1
n
,
1)
⊂
R
.
S
n
∈
Z
+
A
n
=
(0
,
1) = [0
,
1],
but
S
n
∈
Z
+
A
n
=
S
n
∈
Z
+
[
1
n
,
1] = (0
,
1].
17.8) Let
A
,
B
and
A
α
denote subsets of a space
X
. Determine whether the following equations
hold; if an equality fails, determine whether one of the inclusions
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 Spring '10
 derekwise
 Topology, Closed set, limit point, Hausdorff, disjoint neighborhoods

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