What is topology?
Take a look at these two spaces: a sphere and a donut.
Imagine that they are made of clay. Here’s a question for you:
Can you deform the sphere into the torus continuously (i.e: without ripping it into pieces or poking any holes)?
Intuitively, the answer is no. The solid sphere has no holes but the torus has a hole. So there should be no way of deforming the sphere into the torus without “poking a hole” through it.
But how do you phrase this rigorously? This is the subject of a branch of math called topology.
Topology is notoriously difficult subject to self-study. Luckily, there are many resources online to self-study the topic.
In this issue, I’ll provide some learning resources to self-study topology.
How do I learn topology?
On a high level, topology has two main branches: point-set topology and algebraic topology.
Point-set topology deals with the basic set-theoretic constructions that are useful in topology. It is the foundation used for many other forms of topology.
Algebraic topology uses tools from abstract algebra to study topological spaces.
Point Set Topology Learning Recommendations
Online Notes. By far my favorite place to learn Point Set topology are these online notes by Ivan Khatchatourian. They are very well-motivated and have tons of examples to play with.
They come with a nice list of problems (called the Big List of problems) to practice all the concepts. The problems are on the same link as above.
Textbooks. Another nice resource is the book Topology by James Munkres. It is very clearly written and also has lots of motivational examples. It is a bit dry so it might be helpful to rely mostly on the notes above and turn to Munkres when you need a second perspective.
Algebraic Topology Learning Recommendations
Online Lectures. I love these online lectures by Pierre Albin. They explain three core concepts in algebraic topology: the fundamental group, homology and cohomology. The lecturer Pierre Albin is phenomenal at explaining these concepts very clearly.
Textbooks. The bible for learning algebraic topology is the book “Algebraic Topology” by Allen Hatcher. This is a very nicely written book and Hatcher has made it available for free on his website.
Challenge Problem
As always, here’s a challenge problem.
Consider 10 people sitting around a circular table. In how many different ways can they change seats so that each person has a different neighbor to the right?
Got a solution to the challenge problem? Submit it here.
Solution from last week
See here for the solution to last week’s problem. (Special thanks to Mehmet Şamil Çelik from Turkiye, whose solution is being featured this week.)
Thanks for reading and happy learning! Until next time,
Adithya