Alexander trick

Want to extend a homeomorphismPlanetmathPlanetmath of the circle S1 to the whole disk D2?

Let f:S1S1 be a homeomorphism. Then the formula


allows you to define a map F:D2D2 which extends f, for if xS1D2 then ||x||=1 and F(x)=1f(x/1)=f(x). Clearly this map is continuousMathworldPlanetmath, save (maybe) the origin, since this formula is undefined there. Nevertheless this is removable.

To check continuity at the origin use: “A map f is continuous at a point p if and only if for each sequence xnp, f(xn)f(p).

So take a sequence unD2 such that un0 (i.e. which tends to the origin). Then F(un)=||un||f(un/||un||) and since f(un/||un||)0, hence ||un||0 implies F(un)0, that is F is also continuous at the origin.

The same method works for f-1.

In the same vein one can extend homeomorphisms SnSn to Dn+1Dn+1.

Title Alexander trick
Canonical name AlexanderTrick
Date of creation 2013-03-22 15:53:38
Last modified on 2013-03-22 15:53:38
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 7
Author juanman (12619)
Entry type Definition
Classification msc 37E30
Classification msc 57S05
Related topic Homeomorphism