There’s a saying among mathematicians, that a topologist is someone who cannot tell the difference between a coffee cup and a donut. I’ll discuss that in this post, and I’ll suggest implications beyond mathematics.
Usually, when we say this, we are thinking of the donut and the coffee cup as two-dimensional surfaces. Once we go to the three-dimensional objects, nobody denies that the donut has a soft and spongy texture which makes it clearly different from a coffee cup.
Topology
Let’s start with a brief rundown on what is topology. It is a branch of mathematics where we discuss ideas such as continuity, convergence, etc. A classic example of convergence is with the sequence 0.9, 0.99, 0.999, … We can see that the sequence gets closer and closer to 1, and we say that it converges to 1. So topology has something to do with the geometric ideas of getting closer. But it does so without needing a notion of metric (or distance).
The Heretical Philosopher 