Fall 2007
ARE211
Problem Set #02
Second Analysis Problem Set
Due date: Sep 18
Problem 1
For the following problem, consider an arbitrary universe X, together with an arbitrary metric d
defined on
X
×
X
.
a) Prove that the union of arbitrarily many open sets is an open set.
b) Prove that the intersection of finitely many open sets is an open set.
c) Identify what part of your proof in part (b) would no longer hold if you would consider
infinitely many sets.
d) Prove that the intersection of arbitrarily many closed sets is closed. (This is another very
short proof, you just have to come up with a little trick.
Hint: use DeMorgans formula
∪
i
∈
I
(
X
\
A
i
) =
X
\
(
∩
i
∈
I
A
i
) )
(Note: The union of two sets
A, B
is:
A
∪
B
=
{
x
∈
X

x
∈
A
∨
x
∈
B
}
). The intersection of two
sets
A, B
is:
A
∩
B
=
{
x
∈
X

x
∈
A
∧
x
∈
B
}
).
Problem 2
For the following problem, consider an arbitrary universe X, together with an arbitrary metric d
defined on
X
×
X
. Say whether each of the following statements is true or not. If the statements
are true, prove them. If the statements are wrong, give a counterexample.
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 Fall '07
 Simon
 Topology, Empty set, Metric space, accumulation point, Closed set, General topology

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