M. Flashman

More on Open Sets of Real Numbers

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.

**Proposition**: If U and V are open sets, then U
intersect
V is an open set.

**Proof**: Suppose x is in U intersect V. Then x is in U and x
is
in V.

Since x is in U and U is an open set, there are some numbers a and b where x is a member of (a,b) and (a,b) is a subset of U. Since x is in V and V is an open set, there are some numbers c and d where x is a member of (c,d) and (c,d) is a subset of V.

Let p denote the maximum of the numbers a and c and let q be the minimum of the numbers b and d, then x is a member of (p,q) and (p,q) is a subset of U intersect V.

Thus U intersect V is an open set. **EOP.**

- Is the statement in the proposition conditional or absolute? If conditional, what are the hypotheses and conclusions? If absolute, can you rephrase the statement as a conditional statement?
- The proof of this proposition is direct. State the way in which the argument proceeds.(What is assumed? What is actually demonstrated? Does the proof proceed forward or is it mixed with some backward argument? If it has some backward argument, indicate briefly how the original conclusion is altered.)
- Did the proof implicitly or explicitly leave some steps for the reader to complete? If so, state what steps you think the reader is expected to complete. [Optional: complete these steps.]
- Indicate any parts of the argument that you felt needed
greater detail
or better connection to the proof. [Optional: Supply these
details or suggest
a better connection.]

- Prove {x: x > a} and {x
: x < b} are open sets. [This
proof should be directly from the definition.- see Proof
Evaluation #2.]

- Using Problem 5 and the proposition, prove that if a<b then (a,b) is an open set. [Hint: How is (a,b) related to the sets in problem 5?]