Place & Time:
Spring 2020, Tuesday & Thursday 9:30 - 10:45, on Zoom (meeting ID 993-076-018)

Piazza page

Instructor:
Florian Stecker

E-Mail: stecker@utexas.edu

Zoom PMI: 974-027-4742

XMPP: stek@conversations.im

Discord: m17@florianstecker.de

Telegram: +49 163 161 123 1

Office hours: Tuesday 13:00 - 14:00 & Thursday 15:30 - 16:30, on Zoom

TA: Max Riestenberg, RLM 9.116, riestenberg@math.utexas.edu

TA office hours: Wednesday 12:00 - 14:00

Course contents:

- Some set theory
- Topological spaces and continuous maps
- Interior and closure, limit points
- Connectedness
- Compact spaces
- Countability and separation axioms
- Metric spaces and metrization theorems
- Topology of function spaces
- Homotopy and the fundamental group

Schedule (click link to see notes):

Jan 21 | Sets and functions |

Jan 23 | Cardinality |

Jan 28 | Topological space, basis, standard topology on R and R^{n} |

Jan 30 | Subbasis, coarser and finer topologies |

Feb 4 | Product and subspace topology, closed sets |

Feb 6 | Closure and interior, dense subsets, convergent sequences, Hausdorff |

Feb 11 | Continuous maps, homeomorphisms |

Feb 13 | Conectedness, path connectedness, path connected implies connected |

Feb 18 | Local (path) connectedness, connected components, topologist's sine curve |

Feb 20 | Compact spaces: closed subsets are compact, preserved by continuous functions |

Feb 25 | Products of compact spaces are compact, Heine-Borel theorem |

Feb 27 | Proof of Heine-Borel, metric spaces |

Mar 3 | closure, continuity and compactness in metric spaces, countability |

Mar 5 | compactness and convergent subsequence |

Mar 12 | separation axioms |

Mar 31 | Urysohn Lemma (notes, video) |

Apr 2 | Urysohn metrization Theorem (notes, video) |

Apr 7 | Equivalence relation, quotient topology (notes, video) |

Apr 9 | maps from quotients, group actions (notes, video) |

Apr 14 | polygons, labellings, gluings (notes, video) |

Apr 16 | labelling scheme determines quotient, gluing of non-convex polygons (notes, video) |

Apr 21 | glued polygons are compact surfaces (notes, video) |

Apr 23 | The fundamental group (notes, video) |

Apr 28 | Base point independence and functoriality of the fundamental group, simple connectedness (notes, video) |

Apr 30 | The fundamental group of the circle (notes, video) |

May 5 | The fundamental group of a bouquet of circles (notes, video) |

May 7 | The fundamental group of a polygon gluing (notes, video) |

Homework:
I will post an exercise sheet to this website every Thursday. Please **upload your solutions to Canvas** on the following Thursday. If possible, please submit PDFs which were scanned or written on the computer, but we will also accept photos if necessary. You are allowed (and encouraged) to work in groups of two. However, do not copy solutions or look them up on the internet!

Exercise sheet | due date | remarks |

Sheet 1 Solutions | Jan 30 | fixed mistake in #3: the union should include 0 |

Sheet 2 Solutions | Feb 6 | For #1 you might want to wait until Tuesday. |

Sheet 3 Solutions | Feb 13 | |

Sheet 4 Solutions | Feb 20 | |

Sheet 5 Solutions | Mar 3 | added a hint to exercise 1 |

Sheet 6 Solutions | Mar 12 | corrected mistakes in #4 and #5b |

Sheet 7 Solutions | Apr 2 | |

Sheet 8 Solutions | Apr 9 | Apr 5: fixed typo and improved hint |

Sheet 9 Solutions | Apr 16 | |

Sheet 10 Solutions | Apr 23 | Apr 19: Added hint for #4 and conclusion for #3 |

Sheet 11 Solutions | Apr 30 | |

Sheet 12 Solutions | May 7 |

Exams: We will have a midterm and a final. The final will be a take-home exam. You will have 3 hours to work on it, which you can choose as you want.

- Midterm: Tuesday, March 10, 9:30 - 11:00 exam with solutions
- Final exam: 3 hours between Thursday, May 14 and Saturday, May 16. exam with solutions

Grading: The final grade will be a weighted average of homework (30%), midterm (30%) and final exam (40%). The result will be converted into a letter grade using the following table.

85% - 100% | A |

80% - 85% | A- |

75% - 80% | B+ |

70% - 75% | B |

65% - 70% | B- |

60% - 65% | C+ |

55% - 60% | C |

50% - 55% | C- |

0 - 50% | F |