On the definition of topology

Like most people, my first time to know the word of topology is from the context like a coffee mug has the same topology as a donut because they can be deformed “continuously” to each other if made in plasticine. Such example is very intuitive and satisfying.

But when I was a senior in college, I was taught the rigorous definition of topology in a course on differential geometry for physicists. Since then I haven't really understood such rigorous definition and how it relates the picture of mug and donut. The course started with the definition of topology space via open set, followed by some examples like trivial, discrete and usual topology. Although I learned a lot from the course, I was not able to see the mug and donut in the rigorous definition. Later during the graduate study, I tried multiple times to understand some topology, for example, by reading the top rated book “geometry, topology and physics”. My Ph.D. advisor really loves reading this book. Although the book is already much less abstract than mathematics textbooks and contains a lot of physics examples, I am afraid that it is only for smart students to self study. To me, the book is “too dry” and lacks sentences or paragraphs on the intuitions for the definitions. See the pages 81-82 for its introduction to topology space as an example. 

Yesterday I heard a surprising news that Prof. Canbin Liang(梁灿彬) passed away. I read a few pages of his Chinese book “differential geometry and general relativity” before sleep. It is the follow page that makes me understand the definition of topology space:

To describe how a coffee mug is continuously deformed to a donut, the key is to define what a continuous map or deformation is. In calculus, we define the continuous function in the $\epsilon-\delta$ language, which involves the distance between two points. Meanwhile, there is an alternative but equivalent way to define a continuous map. It is in terms of the open set $(a, b)$ and does not reply on the concept of distance. It is this alternative definition of continuous map that motivates the definition of topology, i.e., we define topology in such a way that it is the minimal structure on which we are able to define a continuous map. The definition of topology is the results of many many mathematical thoughts, but was taught to me in a reverse way as the beginning of the story. 

A good math book should contain enough contexts and explanations on the intuitions and motivations for the mathematical definitions, especially for self study. In the last semester of my undergraduate life, Joking and I registered an undergraduate course “differential geometry” in the department of mathematics. I intended to learn some Riemann geometry for general relativity from this course. It turned out that the course is only about curves and surfaces in 3d Euclidean space. Although I did not learn the Riemann tensor directly, this course succeeded in conveying the intrinsic view of geometry and made me understand some basic elements in Riemann geometry. The advanced formalism in terms of metric, Clifford coefficient, covariant derivative, Riemann tensor, etc is the generalization of the basic first and second fundamental forms as well as Gauss-Bonnet theorem in 2d surfaces. The essential ideas are the same. What I want to say is that there exists such an introductory course for mathematics students before learning “advanced” differential geometry. But physics students learning general relativity are usually thrown into the pool of “advanced” math without such preparation. How much students really grasp depends on how smart they are.