(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
mentioned below the standard metric is assumed.
be closed. Show that there is a point
- 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
- Suppose and are connected subsets of a metric space and
show that is connected.
be a finite number of compact sets in a
metric space Show that
be a countable collection of open
subsets of a complete metric space, Suppose that for each
and that for each
- Let be a bounded sequence in
Show that there exists
and a subsequence such that
Richard B. Melrose