The test on Thursday will be open book - just the book, nothing else is permitted (and no notes in your book!) Note that where is mentioned below the standard metric is assumed.

- Let be closed. Show that there is a point such that
- Give a counterexample to each of the following statements:
- Subsets of are either open or closed
- A closed and bounded subset of a metric space is compact.
- In any metric space the complement of a connected set is connected.
- Given a sequence in a metric space, if every subsequence of that sequence itself has a convergent subsequence then the original sequence converges.

- Suppose and are connected subsets of a metric space and that show that is connected.
- Let be a finite number of compact sets in a metric space Show that is compact.
- Let be a countable collection of open subsets of a complete metric space, Suppose that for each and that for each Show that
- Let be a bounded sequence in Show that there exists and a subsequence such that converges absolutely.

Richard B. Melrose 2004-05-18