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Reminder of the Definitions:

(1) For a and b real numbers with a < b, (a,b) ={ x : a < x <
b}

(2) A set of real numbers, O, is called an open set if and only if
for any number x that is a member of O there are some numbers a and b so
that x is a member of (a,b) and (a,b) is a subset of O.

(3) Suppose F is a family of sets. We define the intersection of the family of sets F by

Proposition 1 : {5} is not
an open set.

Proof: Suppose {5} is an
open set.

Consider the number 5, which
is an element (in fact the only element) of {5}. Suppose a and b are any
real numbers, where a < 5 < b. Then a < (5+a)/2 < 5 and therefore
(a,b) is not a subset of {5}. Thus {5} is not an open set.** EOP.**

Proposition 2:
**[This proposition is FALSE.]**
**If F**_{}
is a family and any member of F is open set of real numbers, then Ç F
is an open set.

Proof:* **[This
proof is erroneous.]*

Suppose *x* is a member
of **Ç F
****. Then ** *x* is a member of A for every A in F.
Since **A**_{
}is an open set, there are real
numbers a and b where *x* is a member of (a,b) and (a,b) is a subset
of **A**
for every A
in F, and hence (a,b) is a subset of Ç F_{
}_{. }**Therefore**_{ }**Ç **F_{
}is an open set. EOP.

Proposition 3:

For n a positive integer, **let A**_{n}
= (5-1/n, 5 + 1/n) and let F = {A_{n} : n a positive integer}. Ç
F= {5}.

- Are the statements in the propositions conditional or absolute? If conditional, what are the hypotheses and conclusions? If absolute, can you rephrase the statement as a conditional statement?
- Proposition 2 is false. Why does Proposition 3 show that Proposition 2 is false.
- The proof of Proposition 2 has an error in it. Describe any errors you find in this "proof" of proposition 2.
- Indicate any parts of the argument in proposition 1 that you felt needed greater detail or better connection. [Optional: Supply these detail or suggest a better connection.]
- For n a positive integer,
**let B**_{n}= (4-1/n, 6 + 1/n) and let F = {B_{n}: n a positive integer}.

Prove: Ç F is the closed interval of real numbers [4,6].

- True or false: The intersection of a family of open sets of real numbers is open. Discuss your response.