Course Outline

Introduction

This course is an introduction to topology, the foundation of modern geometry and analysis. The course will be held in English. Core topics include: topological spaces, continuous maps, compactness, and connectedness. Additional topics will be considered as time allows.

Prerequisite

Per the course curriculum, you should have already taken Analysis I through Analysis III (Foundations of Analysis, Infinitesimal Calculus, and Functions of Many Variables). In particular, you should be reasonably comfortable with metric spaces, and have a good level of mathematical maturity and comfort with abstraction.

Instructor:
Prof. Russ Woodroofe
Email:
russ woodroofe at famnit upr si
Office:
Kettejeva 1 II/10
Office hours:
Monday at 14h, or by appointment

Textbook and literature

The textbook for this course will be Topology, by James Munkres, 2nd edition. Some books for further reading are suggested under More Information.

Topics

We'll begin with the definition of a topological space (and of a topology). This definition requires a bit more motivation than most definitions in the undergraduate mathematics curriculum. A main example of a topology is the family of open sets of a metric space.

We'll proceed to abstract many notions that may be familiar to you in metric spaces into arbitrary topological spaces, such as compactness, connectedness, product, etc. We'll also discuss new constructions such as quotient spaces.

As time allows, we'll consider additional topics. Possibilities include Tychanoff's theorem on compact products, the beginnings of algebraic topology, etc.

Course format, exams and grading

Homeworks will be posted weekly to the e-classroom. Homework assignments will comprise 20% of your grade. The remaining 80% of your grade will be determined by exams.

There will be 1 midterm examination (kolokvij), to be held (outside of lecture) at a mutually convenient time midway through the semester. I highly recommend that you study for and take the midterm exam, even if you don't feel quite ready for it.

In accordance with university policy, there will be at least 4 opportunities to take an instance of the final. (However, you may only take the final once more after you have attained a passing grade.)

The exam portion of your grade will be made up either of midterm and final, or else of final alone. More specifically, your overall grade will be determined by

20%(homework) + max( 30%(midterm) + 50%(final), 80%(final) )

You may bring one hand-written A4-sized sheet of paper with you to each exam, but no books, calculators, or other aids.

Midterm and final exams will be in-person.

Collaboration

I expect much of your learning to take place in working out the homework problems. While you may collaborate with other students on the homework, I expect you to have thought hard about the problems on your own first. If you do collaborate substantially, then you should so indicate on your homework paper.
Your write-up should in any case represent your own solutions, written in your own words. Copying the solutions from another student or from an internet source is a form of academic dishonesty, and will not be tolerated.

Obviously, exams will be strictly your own work.

Last modified February 20, 2024