Problem Statement: Let be an uncountable collection of open sets in . Let . Prove there exists a countable collection of open intervals satisfying (1) and (2) for every positive integer there exists an such that .
Proof: Since the rationals are countable we may enumerate the rationals and any subset of the rationals. Let be an enumeration of the rationals which are contained in . Then each for some collection of . Define to be the largest open interval around such that the interval is still completely contained within some .
First note that since each is determined by it follows that is a countable collection of open intervals since we may simply use the enumeration on to enumerate .
Furthermore, by construction it follows that for every positive integer , there exists some such that . Thus property (2) is satisfied.
Now we must show that . As stated above each for some , thus .
Let . We wish to show that for some . If then for some positive integer and so .
If then for some . Create the open interval with radius such that is the largest radius possible for to still remain inside . Now consider the interval . Since is dense in we may find a . Now put an open interval around with radius such that is the largest possible radius such that . Then this interval is contained in by construction. Since we chose it follows that . Thus, and so . Since we have set containment both ways, condition (1) is satisfied.
Therefore, there exists a countable collection of open intervals satisfying the conditions.
Reflection: This was not a straightforward problem. The thing that makes it work though is how we construct the . Originally we constructed them based on which we were in, but then we came up the issue that any rational could be in several since they can overlap and so this meant that we were creating uncoutably many . No bueno. So, we had to come up with a way to fix that. Dr. Retsek had the great idea to define our so that they were as large as possible. Also, the second condition is kind of hinting at that since your must be contained in for some . Also, drawing a picture really helped us get an idea of what was going on here.
When trying to go from something that is uncountable to something that is countable you want to try and find a countable subset of your uncountable thing. The rationals worked really well since they are dense.
On another note, 10 days remaining.