This site contains mathematics courses and book drafts and provides image analysis software. Created and run by Peter Saveliev.

# Topology Illustrated

### From Intelligent Perception

*Topology Illustrated* by Peter Saveliev

**Foreword**

The text follows the content of a fairly typical, two-semester, first course in topology. Some typical topics are missing though: the fundamental group, classification of surfaces, and knots. Point-set topology is presented only as much as seems necessary in order to develop algebraic topology. The focus is *homology*.

The presentations are often more detailed than one normally sees in the textbooks on this subject which makes the text appropriate for self-study. Exercises are numerous. Pictures are in the hundreds. Applications serve as another way to illustrate topological ideas. Some of the topics are: the shape of the universe, configuration spaces, digital image analysis, data analysis, social choice, and, of course, calculus.

The way the ideas are developed may be called "historical" but not in the sense of what actually happened -- it's too messy -- but rather what *ought to have happened*.

All of this makes the book a lot longer than a typical book with a comparable coverage.

A rigorous course in linear algebra is an absolute necessity. In the first half of the book, the need for a modern algebra course may be avoided but not the maturity it requires.

Where the book leaves off, one usually proceeds to cohomology or singular homology.

Work in progress...

Note: Ignore the occasional links as you read the sections below. Instead, click *Back* when you are finished and continue to the next section.

**Chapter 1. Homology classes**

- Topology around us
- Introduction to homology
- Topology of graphs
- Homology groups of graphs
- Maps of graphs

**Chapter 2. Topologies**

- Introduction to point-set topology
- Neighborhoods and topologies
- Topological spaces
- Continuous functions
- Topological equivalence
- Relative topology

**Chapter 3. Complexes**

- Euclidean space with discrete structure
- Chains of cubes
- The chain complex
- Cubical complexes
- Oriented chains
- Data with Euclidean topology
- Simplicial complexes
- Simplicial maps and chain maps

**Chapter 4. Spaces**

**Chapter 5. Maps**

Appendix: Student's guide to proof writing

**References**

These are the books I've used for teaching in the past: