Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. (https://en.wikipedia.org/wiki/Homeomorphism)
A continuous deformation between a coffee mug and a donut (torus) illustrating that they are homeomorphic.
I tried to demonstrate homeomorphic operations about sphere or torus.
Does mathematica or other software have a ready-made package to demonstrate the homeomorphic operation of the surface? It is best to use the mouse to demonstrate the surface arbitrary continuous torsion process. It's like playing with silly putty. The process may be irregular or arbitrary. Note that the change trajectory of the above animation has certain rules. This has some limitations on display.
The production of dynamic layout of graph which was seen as a One-dimensional topological space in graph theory seems to be similar to my idea.
