MAT3333

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:

• 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	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.