Information about the Final Exam for Math 4853-001

Mathematics 4853-001 - Introduction to Topology - Spring 2010

Information about the Final Exam

The Final Exam will be in the usual classroom on Wednesday, May 12, at 1:30 p. m. You may work until 3:45 p. m., if you wish, and should not have difficulty completing whatever you are able to do on the exam in that amount of time. The current draft of the exam has 75 points.

As usual, blank paper for your solutions will be provided, and you may have as much paper as you need and may put the problems in any order. Please hand in your exam paper along with your solutions.

The following topics are the main content of the exam:

  1. Topologies and the topological definition of continuity, homeomorphisms.
  2. Countable and uncountable sets, the classic Cantor arguments.
  3. Bases and the Basis Recognition Theorem.
  4. Products of topological spaces, coordinate functions and the Mapping Into Products Theorem.
  5. Compactness. The definition, basic properties, compactness and continuous maps.
  6. Sequences and convergence. Definitions, proving that sequences do or do not converge. Uniform convergence, the Uniform Convergence Theorem.
  7. Metric spaces, the metric topology, metrics on spaces of functions.
  8. Completeness and complete metric spaces, spaces of continuous functions. This will receive some extra weight since it has not been covered on the in-class exams.

The following will not appear: epsilon-delta proofs, the Mean Value Theorem and its proof, subbases, least upper bound arguments, cocountable topology, the proof that a metric d:X\times X\to R on X is continuous (although this fact itself may be of use on the exam), difficult details of the proofs of the last few theorems done in class.

1. Topologies and the topological definition of continuity, homeomorphisms.
2. Countable and uncountable sets, the classic Cantor arguments.
3. Bases and the Basis Recognition Theorem.
4. Products of topological spaces, coordinate functions and the Mapping Into Products Theorem.
5. Compactness. The definition, basic properties, compactness and continuous maps.
6. Sequences and convergence. Definitions, proving that sequences do or do not converge. Uniform convergence, the Uniform Convergence Theorem.
7. Metric spaces, the metric topology, metrics on spaces of functions.
8. Completeness and complete metric spaces, spaces of continuous functions. This will receive some extra weight since it has not been covered on the in-class exams.