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NUS Math Module Review: MA4266 Topology

Assessment Structure:

10% Tutorial Participation

20% Homework (2 sets)

20% Midterm (In-Class)

50% Final (Online)

This is a module under the MA4 list, allowing Pure Math majors to take this to satisfy their core requirements. The module’s topics include basics and definitions of topological spaces, connectedness and compactness, countability and separation axioms, Tychonoff’s theorem, complete metric spaces and function spaces. Prof. Zhang also spent the last few lectures on an introduction to algebraic topology, which was not tested. The textbook of reference is Topology (Second Edition), James Munkres.

Lecture attendance is not compulsory but tutorial attendance is. Tutorials involve students presenting their solutions in the front, and their tutorial participation grade will depend on their presentation. Prof. Zhang opened a forum for students to indicate their interests to present certain problems, and he ensures that every student is given mostly equal chance of presenting. Tutorial problems are often tested topics which he decides is best left for students to figure out themselves (and hence are not explained in lecture, but may be brought up in lectures after the tutorial). Topics raised include order topology, topological groups (due to COVID-19 situation, last few weeks of tutorials were dropped, so topological groups were no longer tested), and construction of various counterexamples in topology. Tutorial questions are often very difficult, so do not be surprised if you need to seek much online help for them.

Homework is split to 2 sets, and questions from homework are significantly more manageable than that of tutorial questions. Homework questions mainly serve to reinforce concepts taught in lectures (unlike tutorial questions).

This module was really difficult. By generalising metric spaces to general topology, many heuristics become blurred, and many results that are true in metric spaces (from MA3209) become either false or non-trivial results in general topological spaces. To master the concepts in topology, I believe that one must have very strong visualising skills and grasp of the concepts to understand the theorems. Many counterexamples in topology are very artificial – in short, this means that it’s probably impossible for one to come out with such a counterexample without studying the statement for months, or maybe years. Furthermore, after seeing such a counterexample, the first impression is usually “oh, I guess it’s false” instead of “right, no wonder it’s false!”. I always refer the topology as “set theory with analysis”, but ironically, I find topology one of the least intuitive modules of the semester (I mentioned in my previous reviews that both set theory and analysis are intuitive).

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