Difference between revisions of "MAT3333"

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Revision as of 15:13, 25 March 2023

Course name

MAT 3333 Fundamentals of Analysis and Topology.

Catalog entry: MAT 333 Fundamentals of Analysis and Topology. Prerequisite: MAT 3003 Discrete Mathematics, or consent of instructor. Topology of the real line. Introduction to point-set topology.

Prerequisites: MAT 1224 and MAT 3003.

Sample textbooks:


Topics List

(Section numbers refer to Erdman's book.)

Week Sections Topics Student Learning Outcomes

1

1.1. Appendices C, G & H.

Operations, order and intervals of the real line.

  • Arithmetic operations of ℝ.
  • Field axioms.
  • Order of ℝ.

2

1.2. Appendix J.

Completeness of the real line. Suprema and infima.

  • Intervals: open, closed, bounded and unbounded.
  • Upper and lower bounds of subsets of ℝ.
  • Least upper (supremum) and greatest lower (infimum) bound of a subset of ℝ.
  • The Least Upper Bound Axiom (completeness of ℝ).
  • The Archimedean property of ℝ.

3

2.1, 2.2

Basic topological notions in the real line.

  • Distance.
  • Neighborhoods and interior of a set.
  • Open subsets of ℝ.
  • Closed subsets of ℝ.

4

3.1–3.3

Continuous functions on subsets of the real line.

  • Continuity at a point (local continuity).
  • Continuous functions on ℝ (global continuity).
  • Continuous functions on subsets of ℝ.

6

4.1, 4.2

Convergence of real sequences.

  • Sequences in ℝ.
  • Convergent sequences.
  • Algebraic operations on convergent sequences.

7

4.3, 4.4

The Cauchy criterion. Subsequences.

  • Sufficient conditions for convergence. Cauchy criterion.
  • Subsequences.

8

5.1, 5.2

Connectedness and the Intermediate Value Theorem

  • Connected subsets of ℝ.
  • Continuous images of connected sets.
  • The Intermediate Value Theorem.

9

6.1, 6.2, 6.3

Compactness and the Extreme Value Theorem.

  • Compact subsets of the real line.
  • Examples of compact subsets.
  • The Extreme Value Theorem.

11

7.1, 7.2

Limits of real functions.

  • Limit of a real function at a point.
  • Continuity and limits.
  • Arithmetic properties of limits.

9.1-9.3

Metric spaces.

  • Metric spaces. Examples.
  • Equivalent metrics.

10.1-10.3

Interior, closure, and boundary.

  • Accumulation point.
  • Boundary point.
  • Closure.
  • Examples.

11.1-11.2

The topology of metric spaces

  • Open and closed sets.
  • The relative topology.

12.1-12.3

Sequences in metric spaces.

  • Convergent sequences.
  • Sequential characterizations of topological properties.

14.1-14.3

Continuity and limits.

  • Continuous functions between metric spaces.
  • Topological products.
  • Limits.

15.1–15.2

Compact metric spaces.

  • Compactness: definition and elementary properties.
  • The Extreme Value Theorem.

16.2-16.4

Sequential compactness and the Heine-Borel Theorem.

  • Sequential compactness.
  • Conditions equivalent to compactness of a metric space.
  • The Heine-Borel Theorem.