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The grading of final exam will be submitted soon. We want to clarify two things in the meantime.
- Problem 8 had two valid answers. Both are accepted.
- Problem 12 had a typo: "f-1(A) is proper" should have been "f-1(A) is compact". We took any possible confusion arising from this in consideration.
The model answers can be found here.
Basic frameworks: metric spaces and topological spaces; what do these structures represent?
Basis for topology: what are examples of bases? When do they become convenient?
Product topology: what can be packaged in this setting?
Continuous maps: how are they defined in the framework of topological spaces? Do bases simply this?
Topology on a space of continuous maps: what are sensible topologies (with some extra assumptions on domain / target)?
Quotient topology: how can we use this idea?
Connectedness, path-connectedness
Compactness: what are examples and non-examples?
Homotopy, fundamental group: what do they represent? How can we compute the fundamental group of a space? What can be said from the nontriviality of the fundamental group of the circle?
The exam will roughly have two parts: the first part to mainly decide if you pass the course or not, and the second part to mainly decide your score.
To get enough points in the first part, I want you to have basic but precise understanding of the principles behind this course. For example, you want to know how to mathematically model various
concepts, but “pop-sci” style explanation in plain words would not be enough. On the other hand, you do not have to give precise reasoning in this part.
The second part is more advanced, so you are expected to give rigorous formulations and reasoning. However, you do not have to memorize proofs of various theorems. It is more important that, given a mathematical problem and a tool to study it, be able to explain how to use that tool to study such problem, with an illustrating example.
The list of problems for mandatory assignment is here. You need to solve at least one of them to take the final exam. Typeset your solution by LaTeX, and upload the PDF on the Canvas system. If you are taking this course as MAT3500, you can instead prepare your solution by handwriting and upload the scan. The deadline is October 31.
You can find the exercises for each week on the website for the previous year, i.e. on fall 2021 when John Rognes and me did the course. We will do the exercises from the chapters that Makoto has gone through every week. For example this Wednesday the suggested exercises are:
Section 18: Exercises 1, 3, 4, 5, 6, 9, 10, 11, 12, 13.
Section 19: Exercises 1, 2, 3, 4, 6, 7, 8, 10. (Omit questions about box topologies).
There is no lecture on September 1 (Thu) due to Makoto's travel.