# Practice test 1

(Also available as postscript and acrobat file off my web page).

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.

1. Let be closed. Show that there is a point such that
2. Give a counterexample to each of the following statements:
1. Subsets of are either open or closed
2. A closed and bounded subset of a metric space is compact.
3. In any metric space the complement of a connected set is connected.
4. Given a sequence in a metric space, if every subsequence of that sequence itself has a convergent subsequence then the original sequence converges.
3. Suppose and are connected subsets of a metric space and that show that is connected.
4. Let be a finite number of compact sets in a metric space Show that is compact.
5. Let be a countable collection of open subsets of a complete metric space, Suppose that for each and that for each Show that
6. Let be a bounded sequence in Show that there exists and a subsequence such that converges absolutely.

Richard B. Melrose 2004-05-18