To understand this concept, imagine that you have a piece of clay. You can mold the clay into various shapes, such as a sphere, a cube, or a donut. However, no matter how you manipulate the clay, you cannot transform it into a coffee mug without breaking or puncturing it. This is because the coffee mug has a handle, which creates a hole in the object.
In contrast, if you start with a donut, you can continuously mold and reshape it into the form of a coffee mug without breaking it. To do this, you would gradually push the hole in the donut outward until it forms the cylindrical shape of the mug.
Topologists define objects that are topologically equivalent as belonging to the same "genus". In this case, the genus of the donut and the coffee mug is one. Objects with different genuses cannot be transformed into each other without cutting or tearing.
The concept of topological equivalence has various applications in fields such as mathematics, physics, and engineering. It allows researchers and engineers to study the properties and behavior of objects without getting caught up in their specific shapes or details.
