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?

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    $\begingroup$ When I was an undergrad at Cornell, Marshall Cohen used this theorem in a course. He called it the Klee Trick. I think it was part of his discussion of Poincare Duality for pseudomanifolds. My recollection is pretty vague, though. I suppose you might try emailing Marshall. (I don't know if he still checks his Cornell email, which is the only one I see on-line.) I can also look back at my notes from the course (feel free to remind me to do so!). $\endgroup$ – Dan Ramras Jun 13 '17 at 19:15
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    $\begingroup$ Here is Doyle's note. It took me a little while to find it, so I thought I'd spare others the time. $\endgroup$ – Olivier Jun 14 '17 at 0:43
  • $\begingroup$ I think my recollection was wrong, and the Klee trick was not part of Cohen's discussion of pseudomanifolds. See my answer below. $\endgroup$ – Dan Ramras Jun 25 '17 at 17:09

This answer is a correction to my comment above. In these notes 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 $S^n$.

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Klee's trick easily follows from Tietze extension theorem, and there is a generalization when $\mathbb R^n$ is replaced by a locally convex linear space $E$, which is based on Dugundji's extension theorem.

Incidentally, the sets $K$, $K^\prime$ need only be closed. Compactness is not required.

A common use of these results is the unknotting of compact sets in certain locally convex linear spaces, namely those where $E$ is homeomorphic to $E\times E$: any homeomorphism of compact sets extends to an ambient homeomorphism.

Another application is to the problem of extending a metric: if $(X, d)$ is a metric space, $A$ is a closed subset of $X$, and the restriction of $d$ to $A\times A$ is equivalent to a metric $\rho$, then $\rho $ extends to a metric on $X$. Moreover, if $d$ and $\rho$ are complete, then so is the extension of $\rho$. Here "equivalent" means that they induce the same topology.

All this can be found in "Selected topics in infinite-dimensional topology" by Bessaga and Pełczyński by tracing from Proposition II.3.2.

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