Why sphere and torus are not homeomorphic?

Question

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

enter image description here

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?

"Removing a(n embedded) circle from a sphere always splits it into two parts" is nontrivial, FYI: it's the Jordan curve theorem. — , Jun 26 '14 at 06:01

score 10 · Accepted Answer · answered Jun 26 '14 at 05:41

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.

score 6 · Answer 2 · answered Oct 02 '15 at 20:28

6

Intuitively, homeomorphisms do not allow tearing or gluing. This is because they and their inverse need to be continuous :

Gluing breaks injectivity, hence it breaks continuity.
Tearing breaks continuity.

answered Oct 02 '15 at 20:28

Jean-François Gagnon

1,060

score 0 · Answer 3 · answered Oct 02 '15 at 20:32

0

Even ignoring the invertibility issues, your map doesn't turn the torus into a sphere. It turns the torus into a sphere with a wall in it.

answered Oct 02 '15 at 20:32

hunter

29,847

(perhaps you mean the ball and the solid torus?) – hunter Oct 02 '15 at 20:32

Why sphere and torus are not homeomorphic?

3 Answers3

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