MAT3333

Course name

MAT 3333 Fundamentals of Analysis and Topology.

Catalog entry: MAT 3333 Fundamentals of Analysis and Topology. Prerequisite: MAT 3003 Discrete Mathematics, or consent of instructor. Topological notions in the real line and in metric spaces. Convergent sequences. Continuious functions. Connected and compact sets. The Intermediate Value and Extreme Value theorems. Sequential compactness and the Heine-Borel Theorem.

Prerequisites: MAT 1224 and MAT 3003.

Sample textbooks:

• John M. Erdman, A Problems Based Course in Advanced Calculus. Pure and Applied Undergraduate Texts 32, American Mathematical Society (2018). ISBN: 978-1-4704-4246-0.
• Jyh-Haur Teh, Advanced Calculus I. ISBN-13: 979-8704582137.

Topics List

(Section numbers refer to Erdman's book.)

Week	Sections	Topics	Student Learning Outcomes
1-3	Chapters 1 & 2. Appendices C, G & H.	Operations, order and intervals of the real line. Completeness of the real line. Suprema and infima. Basic topological notions in the real line.	Arithmetic operations of ℝ. Field axioms. Order of ℝ. 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. Distance. Neighborhoods and interior of a set. Open subsets of ℝ. Closed subsets of ℝ.
4	Chapter 3	Continuous functions on ℝ.	Continuity at a point (local continuity). Continuous functions on ℝ (global continuity). Continuous functions on subsets of ℝ.
4-5	Chapter 4	Convergence of real sequences. The Cauchy criterion. Subsequences.	Sequences in ℝ. Convergent sequences. Algebraic operations on convergent sequences. Sufficient conditions for convergence. Cauchy criterion. Subsequences.
6	Chapter 5	Connectedness and the Intermediate Value Theorem	Connected subsets of ℝ. Continuous images of connected sets. The Intermediate Value Theorem.
7	Chapter 6	Compactness and the Extreme Value Theorem.	Compact subsets of the real line. Examples of compact subsets. The Extreme Value Theorem.
8	Chapter 7	Limits of real functions.	Limit of a real function at a point. Continuity and limits. Arithmetic properties of limits.
9 & 10	Chapters 9, 10, 11	The topology of metric spaces.	Metric spaces. Examples. Equivalent metrics. Interior, closure, and boundary. Accumulation point. Boundary point. Closure. Open and closed sets. The relative topology.
11	Chapter 12	Sequences in metric spaces.	Convergent sequences. Sequential characterizations of topological properties.
12	Chapter 14	Continuity and limits.	Continuous functions between metric spaces. Topological products. Limits.
13	15.1–15.2	Compact metric spaces.	Compactness: definition and elementary properties. The Extreme Value Theorem.
14	Chapter 16	Sequential compactness and the Heine-Borel Theorem.	Sequential compactness. Conditions equivalent to compactness of a metric space. The Heine-Borel Theorem.
15	Chapter 18	Complete metric spaces	Cauchy sequences in metric spaces. Metric completness. Completeness and compactness.