M367K (53115): Topology I

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:

Schedule (click link to see notes):
Jan 21Sets and functions
Jan 23Cardinality
Jan 28Topological space, basis, standard topology on R and Rn
Jan 30Subbasis, coarser and finer topologies
Feb 4Product and subspace topology, closed sets
Feb 6Closure and interior, dense subsets, convergent sequences, Hausdorff
Feb 11Continuous maps, homeomorphisms
Feb 13Conectedness, path connectedness, path connected implies connected
Feb 18Local (path) connectedness, connected components, topologist's sine curve
Feb 20Compact spaces: closed subsets are compact, preserved by continuous functions
Feb 25Products of compact spaces are compact, Heine-Borel theorem
Feb 27Proof of Heine-Borel, metric spaces
Mar 3closure, continuity and compactness in metric spaces, countability
Mar 5compactness and convergent subsequence
Mar 12separation 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 sheetdue dateremarks
Sheet 1 SolutionsJan 30fixed mistake in #3: the union should include 0
Sheet 2 SolutionsFeb 6For #1 you might want to wait until Tuesday.
Sheet 3 SolutionsFeb 13
Sheet 4 SolutionsFeb 20
Sheet 5 SolutionsMar 3added a hint to exercise 1
Sheet 6 SolutionsMar 12corrected mistakes in #4 and #5b
Sheet 7 SolutionsApr 2
Sheet 8 SolutionsApr 9Apr 5: fixed typo and improved hint
Sheet 9 SolutionsApr 16
Sheet 10 SolutionsApr 23Apr 19: Added hint for #4 and conclusion for #3
Sheet 11 SolutionsApr 30
Sheet 12 SolutionsMay 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.

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