MCS 332 Exam 2 Info

MCS 332 EXAM 2 INFORMATION

In-class test April 19: closed book and notes

TOPICS

Metric spaces
- Metric, metric topology, metric space, metrizable space
- Familiar metrics on Rⁿ
- Uniform metric on R^J
- Comparison of product, uniform, and box topologies on R^J (Theorem 20.4)
- The "little ell two" topology (p. 128)
- Continuity of a function from one metric space to another
- Uniform convergence in a metric space
Calculus
Connectedness
- Separation, connected
- Totally disconnected
- Theorems telling when the following are connected:
  - Unions of connected subspaces
  - Connected spaces with limit points adjoined, closures
  - f(connected space)
  - Products of connected spaces
- Linear continuum
- Intermediate Value Theorem
- Path, path connected
- (Connected) components, path components
Compactness
- Cover, covering, open covering
- Compact
- Finite intersection property, Theorem 26.9
- Limit point compact (= weakly countably compact)
- Sequentially compact
- Locally compact
- One-point compactification
- Theorems telling when the following are compact:
  - Subspaces
  - Subspaces of Rⁿ
  - f(compact)
  - products of spaces
- Extreme Value Theorem
- Uniform Continuity Theorem
- Lebesgue Number Lemma

Take-home test due April 22: open book and notes

You may use your notes, problems solutions, and the two course texts. Be prepared for problems on:

Metric spaces, including subspaces of R^Z₊
Showing a given space is (or is not) connected
Showing a given subspace is compact or limit point compact or ...
Other

Study hint: Look over both assigned and unassigned problems, including problems discussed in class.

Take-home exam, page 1

Exam 2, Part 3: student-requested problems

1. If the topologist's sine curve was on a baseball team, what position would it play, and why?
2. Do you know anyone who is: --Metrizable? --Compact? --Totally disconnected? Give examples if possible.
3. Perform a one-point compactification on your dog. Submit a final report on the results.
4. Is St. Peter locally compact? Why or why not?
5. Prove that a gondola is topologically equivalent to a dulcimer.
6. Explain why Robert Frost should have called his poem, "Crucial Choices in a Path Connected Subspace at a point x instead of "The Road Less Traveled", and speculate on the literary implications of this change on post-modern culture.
7. Prove that the solution to world hunger is a special case of the Lebesgue Number Lemma.