I heard that two objects are homeomorphic if one could be deformed into the other by continuous transformation. For example in this link, it is shown
a sphere and a torus are not homeomorphic
"Proof"
Removing a circle from a sphere always splits it into two parts — not so for the torus.
However, I may imagine the following operations
to let the points around the inner circle of the continuous torus merge to a sphere. I see no reason merging is not continuous. Why not this transformation follow the definition of homeomorphic?
Best Answer
Because when you merge many points into a single one, you do not have a bijection; a homeomorphism is a continuous map with a continuous inverse, and a non-bijective map cannot have a (two-sided) inverse.
Besides, if this operation was a homeomorphism, then its inverse --tearing a circle to turn it into a torus would be a homeomorphism.