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	Section 1.1. Appendices C, G & H.	Operations, order and intervals of the real line.	Arithmetic operations of ℝ. Field axioms. Order of ℝ. Intervals: open, closed, bounded and unbounded.
2	Appendix J.	Completeness of the real line. Suprema and infima.	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	1.2, 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.