In his *"Some topological properties..." (1955)*, Klee gave a construction (simple and beautiful) of an isotopy $h\_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ in the coordinate $\mathbb{R}^n$-subspace to any other homeomorphic compact $K'$ set in this subspace. 


The idea is to extend the homeomorphisms $K\to K'$ and $K'\to K$ to continuous functions $f,g\colon\mathbb{R}^n\to\mathbb{R}^n$ and construct the needed isotopy as concatenation of $(x,y)\mapsto (x,y+t\cdot f(x))$ and $(x,y)\mapsto (x-t\cdot g(y),y)$ 


Later in *"Plane separation" (1968)*, Doyle used this idea to give 5 line proof of the Jordan separation theorem.



> 
> Do you know any other places where this idea shows up?
> 
> 
> 



Klee's trick --- more applications
This answer is a correction to my comment above. In [these notes](http://math.iupui.edu/%7Edramras/M-Cohen-Klee-trick-notes.pdf) from Marshall Cohen's course (from Spring 2001), he uses the Klee trick to give a simple proof that if $f, g: X \to S^n$ are non-surjective embeddings with $X$ compact, then the integral homology of $S^n - f(X)$ is the same as that of $S^n - g(X)$. From this he gave simple deductions of Invariance of Domain (showing that $\mathbb{R^k}$ does not embed into $\mathbb{R^n}$ if $k>n$ and the Jordan-Brouwer separation theorem for embeddings of $S^{n-1}$ into $S^n$. 


