The Final Exam will be in room 1105 from 9:00 to 11:15 a. m. on Monday, December 13, 2004. The exam is closed-book and closed-notes; that is, all you will need is something with which to write.

It will be similar to the in-class exams. Most of the emphasis will be on the second half of the course, especially the material covered after Exam II. As usual, you can and should make use standard results (for example, a continuous image of a connected space is connected) in your arguments, but not in such a way as to trivialize a problem (for example, if the exam problem is to prove that a closed set contains all of its limit points, one should not just say that since the closure of A is A \cup A', A' is contained in A).

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 concepts such as locally compact, locally path-connected, separable, second-countable, diameter, Lebesgue number, compactification, isometry, etc. It is also necessary to know the exact statements (not word-for-word, but with all essential hypotheses and content) of the major theorems.

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

1. bases and the Basis Recognition Theorem

2. metric spaces and metric topologies

3. path-connectedness and local path-connectedness

4. local compactness

5. the product topology for products of finitely or infinitely many spaces, the fundamental theorem for mappings into products

7. open maps and closed maps

8. the Tietze Extension Theorem

9. the Tychonoff Theorem

10. the quotient topology and quotient maps, the universal property of quotient maps, identification spaces

There will be very little about manifolds on this exam.

The following topics will not appear on this exam:

1. definition of compactness using the finite intersection property

2. uniform continuity

3. uniform convergence

4. one-point compactification

5. path-connectedness of the n-sphere

6. the topologist's sine curve

7. the Intermediate Value Theorem

8. isometries and affine maps of the plane, barycentric coordinates, dilations

9. regular spaces

10. stereographic projection

11. the abstract definition of products of sets

12. Lebesgue numbers