Section B.9 Problem Sheet 9
Due 29 November at 3pm, either in class or in the pigeon hole in 4W.
As usual, I would encourage you to think about each of these questions for at least a bit before peeking at the hint!
Exercise 4.3.1 asks you to show, using the definition, that a very simple metric space is not compact. This is close to the easiest question I can come up with about Definition 4.16.
Exercise 4.3.3 is a classic application of compactness as a way to upgrade ‘local’ properties (like continuity) to more ‘global’ counterparts (like uniform continuity). Exercise 4.3.4 is another exercise about continuous mappings and compactness. It asks you to prove the same result in two different ways: using sequential compactness and using compactness. In addition to just building intuition about how to work with these definitions, this gives you an opportunity to think about which of these equivalent definitions you feel more comfortable with and why.
Exercise 4.4.1 gives a useful characterisation of relative compactness in terms of subsequences, which is often more relevant in applications. This point of view will be useful to have in mind during Section 4.5.
Finally, Exercise 4.5.1 gets us to dip our toes into Section 4.5. It is close to the easiest question I can come up with about Definition 4.28.
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.
