Information about Exam III for Math 4853-001

Mathematics 4853-001 - Introduction to Topology - Spring 2010

Information about Exam III

Exam III will be in the usual classroom on Friday, April 30, 2010. It will cover the product topology, closed sets and limit points, compactness, connectedness, metric spaces and Hausdorff spaces, and sequence and convergence. The current draft has 54 points.

As on any exam, it is wise to start with the problems that you feel more confident that you know how to do, before moving on to others.

Blank paper will be provided, so all you will need is something to write with. Please write your solutions on the blank paper (you may have as many sheets 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 likely to appear, although the exam is not limited to them:

  1. The product topology. The Mapping Into Products Theorem.
  2. Closed sets and limit points, the closure of a subset. Definitions and basic properties.
  3. Compactness. Definition and examples. Know the proof that continuous real-valued functions on compact spaces are bounded. Be familiar with other general properties of compact spaces.
  4. Connectedness. Definition and equivalent conditions, examples. Be aware that in arguments about connected spaces, either of the other two equivalent conditions to the definition may also be used freely as a criterion for connectedness--- that is, any of them may be taken as the definition.
  5. Metric spaces, the metric topology, examples. Hausdorff spaces. Metric spaces are Hausdorff.
  6. Sequences and convergence, definition of convergence, relation with closure, the Sequence Lemma, convergence and compactness.

The following will not appear: metrics on spaces of functions (although these will be covered on the final exam), the Mean Value Theorem and its proof, 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)

1. The product topology. The Mapping Into Products Theorem.
2. Closed sets and limit points, the closure of a subset. Definitions and basic properties.
3. Compactness. Definition and examples. Know the proof that continuous real-valued functions on compact spaces are bounded. Be familiar with other general properties of compact spaces.
4. Connectedness. Definition and equivalent conditions, examples. Be aware that in arguments about connected spaces, either of the other two equivalent conditions to the definition may also be used freely as a criterion for connectedness--- that is, any of them may be taken as the definition.
5. Metric spaces, the metric topology, examples. Hausdorff spaces. Metric spaces are Hausdorff.
6. Sequences and convergence, definition of convergence, relation with closure, the Sequence Lemma, convergence and compactness.