Section B.5 Problem Sheet 5
Due 1 November at 3pm, either in class or in the pigeon hole in 4W.
Exercise 2.1.1 continues our series of exercises on product spaces, and gives you come basic practice working with the definition of density. We used Exercise 2.1.2 in our proof of Lemma 2.4, but it’s also a good exercise in Analysis 1 reasoning with \(\varepsilon\)’s, \(\delta\)’s and \(N\)’s. Exercise 2.1.3 gets you thinking about continuous extensions (and Lemma 2.4!) in a relatively concrete setting.
Exercise 2.2.2 is needed in the proof of Theorem 2.10, and should be relatively quick. It’s so short, in fact, that I’m sneaking in a fifth problem, Exercise 2.2.3. This is an extremely useful tool for finding concrete completions of some familiar metric spaces.
As usual, if you’re looking for additional practice, I’d encourage you to go back and attempt some of the unassigned exercises and turn them in along with the assigned problems.
Please feel free to email me, or drop by office hours (Tuesdays 1:15–2:05 in 4W 1.12), with any questions about these problems whatsoever. I am also more than happy to meet one-on-one or in a small group.
