Last updated Thu Nov 15 23:00 EST 2018
Due-date: Monday, 11/19/18.
You are required to do all of the problems below. You will not be required to hand them all in. I have indicated below which ones you do have to hand in on the due-date. Don't make the mistake of thinking that I'm collecting only the problems I think are important.
The "due date" above is the date that your written-up problems should be handed in, but don't wait to get started on the assignment. You should always get started on problems as soon as we cover the relevant material in class.
A: Rosenlicht Chap. III/ 26 (prove your answer), 32, 33. Suggestion for simplifying #32: prove that the union of two compact subsets is compact, and use induction. In the hint in #33, replace the phrase "cluster point of their centers" with "accumulation point of the sequence formed by their centers". Hand in all three of these problems.
In the context of #33, note that if \(\epsilon\) is as indicated, and \(p,q\) are any points in \(E\) such that \(d(p,q)\leq\epsilon\), then there is some \(i\in I\) such that both \(p\) and \(q\) lie in \(U_i\) (since the closed ball of radius \(\epsilon\) centered at \(p\) contains both \(p\) and \(q\)). In fact, an equivalent statement of #33 is that, under the same hypotheses, there exists \(\epsilon>0\) such that for any points \(p,q\in E\) with \(d(p,q)\leq \epsilon\), there is some \(i\in I\) for which both \(p\) and \(q\) lie in \(U_i\).B: Click here for non-book problems. Of these, hand in only B4b,c,d,e. (You may assume the result of B4a.)
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