HW4
1. Let
X
be an uncountable set and
R
°
X
be the family of all sets which
either countable or their complements (in
X
) are countable. Prove that
R
is a ring.
2. Describe the minimal ring generated by
F
in the following cases:
Describe the minimal ring generated by
F
in the following cases:
(i) Let
F
be a family consisting of only one nonempty set
E
:
F
=
f
E
g
.
(ii) Let
F
be the family consisting of all sets which contain exactly two
points.
3. Let
H
be a family of sets such that
(I)
A; B
2
H
)
A
\
B
2
H
:
(II) If
A; B
2
H
and
A
°
B
, then there there is a °nite number of sets
C
0
; C
1
; :::C
n
°
1
; C
n
such that
(a)
C
0
=
A
°
C
1
°
C
2
°
:::
°
C
n
°
1
°
C
n
=
B
.
(b)
C
j
n
C
j
°
1
2
H
for every
j
= 1
;
2
; :::; n
.
Prove that a family of sets is a semiring if and only if
H
satis°es the
properties (I) and (II).
4. (a) Let
E
be a family of sets and
R
°
(
E
)
be the minimal
°
ring gen
erated by
E
.
Let
b
R
be the union of all minimal
°
rings generated by
countable subfamilies of sets of the family of sets
E
.
Then
b
R
=
R
°
(
E
)
.
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 Spring '08
 Ruan
 Sets, Metric space, Open set, Topological space, Countable set

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