Alexander trick

Want to extend a homeomorphism of the circle $S^1$ to the whole disk $D^2$?

Let $f:S^1\to S^1$ be a homeomorphism. Then the formula

$$F(x)=||x||f(x/||x||)$$

allows you to define a map $F:D^2\to D^2$which extends $f$, for if $x\in S^1\subset D^2$ then $||x||=1$ and $F(x)=1\cdot f(x/1)=f(x)$. 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 $f$ is continuous at a point $p$ if and only if for each sequence $x_n\mathrm{\to }p$, $f\mathit{}\mathrm{(}{x}_n\mathrm{)}\mathrm{\to }f\mathit{}\mathrm{(}p\mathrm{)}$”.

So take a sequence $u_n\in D^2$ such that $u_n\to 0$ (i.e. which tends to the origin). Then $F(u_n)=||u_n||f(u_n/||u_n||)$ and since $f(u_n/||u_n||)\ne 0$, hence $||u_n||\to 0$ implies $F(u_n)\to 0$, that is $F$ is also continuous at the origin.

The same method works for $f^{-1}$.

In the same vein one can extend homeomorphisms $S^n\to S^n$ to $D^{n+1}\to D^{n+1}$.