# Math 472: Introduction to Topology

## Colorado State University, Fall 2016

**Instructor:** Henry Adams
**Email:** henry dot adams at colostate dot edu
**Office:** Weber 125
**Office Hours:**

- Wednesdays 12:50-1:50 in Weber 125,
- or by appointment.

**Lectures:** MWF 12:00-12:50pm in Engineering E204
**Textbook:** *Basic Topology* by M. A. Armstrong.
This book is freely available as a PDF to CSU students if you login on the CSU library webpage.

**Overview:** This course is an introduction to topology. Topics covered include Euler's theorem, topologies on sets, continuous functions, homeomorphisms, sequences and convergence, metric spaces, compactness, connectedness, path-connectedness, identification spaces, homotopy equivalences, the fundamental group, Brouwer's fixed-point theorem, classification of surfaces.

**Syllabus:** Here is the course syllabus.

## Homework

Homework 1 (LaTeX Source) is due Friday, August 26.Homework 2 (LaTeX source) is due Friday, September 2.

Homework 3 (LaTeX source) is due Friday, September 9.

Homework 4 (LaTeX source) is due Friday, September 16.

Homework 5 (LaTeX source) is due Friday, September 23.

Homework 6 (LaTeX source) is due Friday, October 7.

Homework 7 (LaTeX source) is due Friday, October 14.

Homework 8 (LaTeX source) is due Friday, October 21.

Homework 9 (LaTeX source) is due Friday, October 28.

Homework 10 (LaTeX source) is due Monday, November 14.

Homework 11 (LaTeX source) is due Friday, November 18.

Homework 12 (LaTeX source) is due Friday, December 2.

We will have weekly homework assignments. All homework is due in class at the beginning of class. Your homework should be legible and stapled.

## Exams

The exams will be in-class. You will only be able to use your brain and a pen or pencil - no notes, books, or electronic devices. The exams will be comprehensive, except that Midterm 2 will emphasize the material after Midterm 1, and the Final will emphasize the material after Midterm 2. Here is Practice Midterm 1.

Here is Practice Midterm 2.

Here is a Practice Final.

## Notes

Scans of Henry's lecture notes.## Schedule

Date |
Class Topic |
Remark |

Aug 22 | Introduction and course overview | |

Aug 24 | §1.1 Euler's theorem | |

Aug 26 | §1.2 Topological equivalence | Homework 1 due |

Aug 29 | §1.3 Surfaces, §1.4 Abstract spaces | |

Aug 31 | §1.4 Abstract spaces, §1.5 A classification theorem | |

Sept 2 | §1.6 Topological invariants | Homework 2 due |

Sept 5 | Holiday - no class! | |

Sept 7 | §1.6 Topological invariants, §2.1 Open and closed sets | Last day to drop or change grading option |

Sept 9 | §2.1 Open and closed sets | Homework 3 due |

Sept 12 | §2.2 Continuous functions | |

Sept 14 | §3.1 Closed bounded subsets of Euclidean space | |

Sept 16 | §3.2 The Heine-Borel theorem | Homework 4 due |

Sept 19 | §3.3 Properties of compact spaces | |

Sept 21 | §3.3 Properties of compact spaces, §3.4 Product spaces | |

Sept 23 | §3.4 Product spaces | Homework 5 due |

Sept 26 | §3.4 Product spaces | |

Sept 28 | Review | |

Sept 30 | Midterm #1 | Midterm through §3.3 |

Oct 3 | §3.4 Product spaces, 3.5 Connectedness | |

Oct 5 | §3.5 Connectedness | |

Oct 7 | Midterm recap and §3.5 Connectedness | Homework 6 due |

Oct 10 | Research talk and office hours | |

Oct 12 | §3.5 Connectedness | |

Oct 14 | §3.6 Joining points by paths | Homework 7 due |

Oct 17 | §5.1 Homotopic maps | End of course withdrawal period |

Oct 19 | §5.1 Homotopic maps | |

Oct 21 | §5.1 Homotopic maps | Homework 8 due |

Oct 24 | §5.2 Construction of the fundamental group | |

Oct 26 | §5.2 Construction of the fundamental group | |

Oct 28 | §5.2 Construction of the fundamental group | Homework 9 due |

Oct 31 | §5.3 Calculations | |

Nov 2 | Review | |

Nov 4 | Midterm #2 | Midterm through §5.1 |

Nov 7 | §5.3 Calculations | |

Nov 9 | §5.3 Calculations | |

Nov 11 | Class cancelled | |

Nov 14 | §5.4 Homotopy type | Homework 10 due |

Nov 16 | §5.4 Homotopy type | |

Nov 18 | §5.4 Homotopy type | Homework 11 due |

Fall Recess, Nov 21-25 | ||

Nov 28 | §2.3 A space-filling curve | |

Nov 30 | §5.5 The Brouwer fixed-point theorem | |

Dec 2 | §5.7 The boundary of a surface | Homework 12 due |

Dec 5 | An introduction to computational topology | |

Dec 6 | Dinner at Henry's! | |

Dec 7 | Review and class picture | |

Dec 9 | Review | |

Final Exam, Monday December 12 |
||

Through §5.7 | ||

4:10-6:10pm in Engineering E204 |