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.)

5 6 7

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.
---	Review. First midterm exam.
4.1, 4.2	Convergence of real sequences.	Sequences in ℝ. Convergent sequences. Algebraic operations on convergent sequences.
4.3, 4.4	The Cauchy criterion. Subsequences.	Sufficient conditions for convergence. Cauchy criterion. Subsequences.