MCS 332 Exam 3 Info

MCS 332 EXAM 3 INFORMATION

Take-home exam given out May 17, due May 20: open book and notes

Preparation: Review assigned homework problems and problems worked in class. Also look at some other problems in sections we have covered. Prepare for problems relating to the countability axioms, the separation axioms, the Tychonoff theorem, and complete metric spaces and function spaces. Make notes on sufficient conditions or NASCs for a property to hold--conditions that may be useful in constructing proofs. Be prepared to cope with novel problems.

Instruction page for take-home part of Exam 3

In-class exam May 20: closed book and notes

TOPICS

Countability and separation axioms
- Hereditary properties
- G_delta sets
- Countability axioms
  - First-countable
  - Second-countable
  - Lindelof
  - Separable
  - Implications between these properties
  - Examples
- Separation axioms
  - T₀-T₄
  - Hausdorff, regular, normal
  - Implications between these properties
  - Examples
- The Urysohn Lemma
The Tychonoff theorem
- (Use of) Zorn's lemma
Complete metric spaces and function spaces
- Cauchy sequence, complete metric space
- Uniform metric
- Sup metric
- Completeness of Euclidean space in familiar metrics
- Sufficient conditions for Y^X, C(X,Y), B(X,Y) to be complete in the uniform metric
- Contraction
- Contraction mapping principle (Exercise #43.5)
- A space-filling curve
- Totally bounded
- Pointwise bounded
- Equicontinuous
- NASC for a metric space to be compact
- The classical Ascoli theorem