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