# Extending homeomorphisms between compact metric subsets

Let $$X$$ be a compact metric, second countable space with finite covering dimension. Let $$A,B$$ be two closed subsets of $$X$$. Assume that $$h:A\to B$$ is a homeomorphism. Is it possible to extend $$h$$ to a homeomorphism $$\tilde{h}: X\to X$$?

If not, are there at least some conditions under which it is possible?

Thanks!

• Let $X=[-1,1]$, let $A=\{0\}$ and let $B=\{1\}$. There is no homeomorphisms of $X$ that maps $0$ into $1$. – erz Oct 6 '18 at 5:57
• If $X$ is homogenous, you can do it for two singletons. – Henno Brandsma Oct 6 '18 at 8:57
• Knot theory deals with waysto embed $S^1$ into $\mathbb{R}^3$. Such embeddings can be distinguished by their complement. So there are plenty of knots where their complements are not homeomorphic. Thus there cannot be a homeomorphism as in this question. – HenrikRüping Oct 6 '18 at 13:55

In the "positive" direction, there exist some deep results (called Z-set unknotting theorems) on extensions of homeomorphisms between Z-sets of Menger manifolds, see Theorem 4.1.18 in the book "Inverse Spectra" of Chigogidze.

In the simplest form the Z-set Unknotting Theorem (proved by Bestvina) says that a homeomorphism $$h:A\to B$$ between two $$Z$$-sets of the $$n$$-dimensional Menger cube $$M$$ extends to a homeomorphism of $$M$$.

A closed subset $$A$$ of a compact topological space $$X$$ is called a $$Z$$-set in $$M$$ if the set of maps $$M\to M\setminus A$$ is dense in the space of all self-maps of $$M$$, endowed with the compact-open topology. Each Z-set is nowhere dense in $$M$$.

If $$M$$ is zero-dimensional, then a closed subset $$A\subset M$$ is a $$Z$$-set if and only if it is nowhere dense. In particular, any homeomorphism between closed nowhere dense subsets of the Cantor set $$M$$ extends to a homeomorphism of $$M$$.

As you can see from the comments the answer is: hardly ever.

As mentioned above the case of one-point sets necessitates the space being homogeneous. But that is not enough, say in $$\mathbb{R}$$ when you map $$\{1,2,3\}$$ to itself by interchanging $$1$$ and $$2$$.

You may also want to have a look at Antoine's necklace.