Want to extend a homeomorphism of the circle to the whole disk ?
Let be a homeomorphism. Then the formula
allows you to define a map which extends , for if then and . Clearly this map is continuous, save (maybe) the origin, since this formula is undefined there. Nevertheless this is removable.
To check continuity at the origin use: “A map is continuous at a point if and only if for each sequence , ”.
So take a sequence such that (i.e. which tends to the origin). Then and since , hence implies , that is is also continuous at the origin.
The same method works for .
In the same vein one can extend homeomorphisms to .
|Date of creation||2013-03-22 15:53:38|
|Last modified on||2013-03-22 15:53:38|
|Last modified by||juanman (12619)|