Topology

Topology is the natural evolution of analysis to a more general level. Whereas analysis was concerned with the implications of continuity on the familiar spaces of Rⁿ and Cⁿ, topology seeks to explore all the types of mathematical structures where analytical concepts can be applied in some form. From the loose structure of point-set topology to the requirement that objects admit some type of smooth structure in differential topology, this division studies the implications of applying analysis to all the types of objects where it makes sense to define some sort of analytical structure. Note that while topology originally evolved from the study of partial differential equations by Poincaré, the modern student need only bring a knowledge of limits and continuity from a good calculus course with them, as well as a basic understanding of what a group is.

Course Goal

The goal of this course is to introduce and build proficiency with the basic machinery of modern topology using examples from an area of topology that is well understood, such as the topology of surfaces. The student will then be well-prepared to encounter the strangeness and pitfalls of more abstract and higher-dimensional topology.

Background

Lessons

Lesson 1 What is a topology?
Lesson 2 Continuous maps
Lesson 3 Metric spaces
Lesson 4 Limits
Lesson 5 Directed Sets & Nets

Resources

Universal property of topological sum in slightly sharper form

Offsite Learning Materials

The Geometry Junkyard's Geometric (Surface) Topology Collection
Topology Atlas
Topology Without Tears an accessible introduction to the basics of topology.
Journal of Geometry and Topology - By the end of this course, the student should be able to at the least read the simplest of these papers without their head exploding. ;-)

Courses

An Introduction to Functional Analysis, by John Cagnol, Anna Rozanova-Pierrat, École Centrale Paris
Introduction to Functional Analysis, Richard Melrose, Massachusetts Institute of Technology, Spring 2009.
Math 109a: Introduction to Geometry and Topology, Alexander Gorodnik, IIT Guwahati, 2014 – 2015.
Metric and Topological Spaces, John O'Connor, University of St Andrews, Scotland, 2004.
MATH 4530 : Introduction to Topology , Tomoo Matsumura, Cornell University, Fall 2010.
MATH 4181: Introduction to Topology, David C. Royster, University of Kentucky, Fall 1999.
Math 429: Introduction to Topology, Aissa Wade, Penn State University, Fall 2005.

Ebooks

Topology, Tai-Danae Bradley, Tyler Bryson, and John Terilla, MIT Press 2020. A graduate-level textbook that presents basic topology from the perspective of category theory.