The mathematics of space and continuous transformations

Topology is a fascinating and broad field of mathematics, and its concepts can seem abstract if you’re not familiar with them. To illustrate topology, one often-used example is the Rubber-Sheet Geometry analogy.

Let’s imagine three shapes: a coffee mug, a donut (torus), and a soccer ball. In the eyes of a topologist, the coffee mug and the donut are equivalent because they each have one hole. In contrast, the soccer ball is different as it has no hole. The essence of topology is not about exact measurements such as length, angle, or area, but about the properties that remain unchanged under stretching, bending, or twisting – what topologists refer to as “continuous transformations.”

If you consider that the coffee mug is made of a flexible material such as rubber, you could imagine deforming it into a donut shape without tearing or gluing it. The handle of the mug represents the hole in...