Exam II will be in the usual classroom at 7:00 p. m. on Tuesday, November 9, 2004. The exam is closed-book and closed-notes; that is, all you will need is something with which to write. You will have up to 2 hours to work.

Try to give clear, concise responses to the questions. You can and should make use of standard results when they can be applied, for example, the Basis Recognition Theorem, the fact that x is in the closure of A if and only if every neighborhood of x meets A, or the fact that every subspace of a Hausdorff space is Hausdorff. It may be helpful to include pictures and diagrams in some of your arguments; of course, a picture does not replace an argument, it just makes it easier for the reader to understand.

For problems with the instruction ``Prove or give a counterexample'', one should give a proof is the statement is true, and give an explicit counterexample if it is false.

It is necessary to know the precise definitions of topological concepts such as locally compact, locally path-connected, separable, diameter, Lebesgue number, compactification, etc.

The following topics will definitely be covered, although the exam is not limited to these topics:

1. separability
2. compactness
3. the product topology for products of finitely many spaces
4. Lebesgue numbers
5. connectedness, the Intermediate Value Theorem
6. path-connectedness and local path-connectedness
7. local compactness
8. the one-point compactification
9. stereographic projection (understand it geometrically, but it is not necessary to know the exact formulas)

The following topics will not appear on this exam:

1. definition of compactness using the finite intersection property
2. open maps, closed maps, local homeomorphisms
3. uniform continuity
4. path-connectedness of the n-sphere
5. the topologist's sine curve
6. surfaces
7. isometries, contraction or expansion mappings
8. products of infinitely many spaces, and any material covered after them