Last updated Fri Nov 22 18:18 EST 2019
"Virtual due-date": Thursday 11/21/19 (the day before the midterm).
Hand-in due-date: Monday 11/25/19.I have indicated below which problems to hand in on Monday 11/25/19. I'm not having you hand in problems Monday to be mean; I'm doing it to give you an opportunity to boost your homework grade.
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". I also suggest making use of another metric-space property that we recently proved is implied by compactness. Of these, hand in only 26.
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 B4d,e.
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