TITLE: Is there a classification of open subsets of euclidean space up to homeomorphism?
QUESTION [24 upvotes]: I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among other such sets or there is an explanation of why such classification is difficult. partial results, like the well known fact that every two open connected and simply connected subsets of the plane are homeomorphic, would be interesting too. thanks.

REPLY [4 votes]: Here's a descriptive set-theoretic view of the problem of classifying open subsets of $\mathbb{R}^2$ up to homeomorphism.  The eventual answer will be a formal version of "it's pretty darn hard, but maybe not inconceivably hard."  Note that I haven't actually done any work here besides placing the work of others in the correct order.
First, a couple of words on classification problems in general (read: propaganda).  When somebody has classified some objects $\mathcal{A}$ in terms of some objects $\mathcal{B}$, two natural questions to ask are
(1) How hard is it to compute the particular $B \in \mathcal{B}$ corresponding to a given $A \in \mathcal{A}$?, and
(2) How hard is it to distinguish elements of $\mathcal{B}$ from each other?
If either of these tasks is particularly hard, one could argue that the classification isn't all that useful.  For example, in the current problem one could say "invoke the axiom of choice to wellorder the quotient space of open sets modulo homeomorphism and classify an open set by the location of its equivalence class in the order," which makes (1) rather challenging.  Alternatively, one could simply beg the question by saying each open set is in a unique homeomorphism-class, so the homeomorphism-classes themselves provide a classification, which makes (1) easy (in a sense) but there's no reason to think (2) can be done simply.
One way to formalize these concepts is through Borel reducibility of equivalence relations.  If $E$ and $F$ are equivalence relations on standard Borel spaces $X$ and $Y$ (resp.), we say $E$ is Borel reducible to $F$, written $E \mathrel{\leq_B} F$, iff there is a Borel function $\phi: X \to Y$ such that $x_0 \mathrel{E} x_1 \iff \phi(x_0) \mathrel{F} \phi(x_1)$.  Loosely, the relation $E \mathrel{\leq_B} F$ means "any definable classification of $Y/F$ is also a definable classification of $X/E$" (or, put more succinctly, "$F$ is at least as complicated as $E$").  This gives us a meaningful way to compare various classification problems.(*)
The $\leq_B$ relation is well studied (but there's still a lot we don't know!) in the case that $E$ and $F$ are in themselves in some sense definable.  Of particular note is when they are orbit equivalence relations induced by a continuous action of a Polish group on a Polish space.  Of even more particular note is when this Polish group is $S_\infty$, the group of permutations of the natural numbers (including those of infinite support).  This group is especially important from a logical point of view.  Given some countable (let's say relational) language $\mathcal{L}$, you can look at the $\mathcal{L}$-structures with universe $\mathbb{N}$.  Then the group of automorphisms of $\mathbb{N}$ respecting $\mathcal{L}$ is a closed subgroup of $S_\infty$.  In particular, isomorphism of $\mathcal{L}$-structures with universe $\mathbb{N}$ is Borel reducible to to an equivalence relation induced by a nice action of $S_\infty$.
Moreover, there's an $S_\infty$-universal equivalence relation $E_{S_\infty}$ in the sense that every equivalence relation induced by a nice $S_\infty$ action Borel reduces to $E_{S_\infty}$.  In a precise sense, $E_{S_\infty}$ is at least as complicated as the isomorphism relation for any sort of countable structure in a countable language (for example, isomorphism of countable graphs, isomorphism of countable groups, etc.).
Now, the punchline of all this is: there's a Borel reduction of $E_{S_\infty}$ to the homeomorphism relation on open subsets of $\mathbb{R}^2$ (equipped with the natural interpretation of the Effros Borel structure).  This follows from 
Camerlo and Gao, "The completeness of the isomorphism relation for countable Boolean algebras," Trans AMS 353 (2001), no. 2, 491-518
and the the pitch-perfect response by André Henriques to a rather nebulous question of mine in the comments after his answer.  The idea is that there is a Borel reduction of $E_{S_\infty}$ to homeomorphism of closed subsets of Cantor space (again with the corresponding Effros Borel structure), constructed in Camerlo-Gao.  Fixing some copy of Cantor space in the plane, this reduction can then be composed with the Borel map $\phi$ that assigns to each closed subset $K$ of Cantor space its complement in $\mathbb{R}^2$.  Since it is known from results of Richards (thanks, Agol!) that $\phi(K_0)$ and $\phi(K_1)$ are homeomorphic iff $K_0$ and $K_1$ are, this composition is indeed a Borel reduction.  In fact, if you restrict your attention to nowhere dense subsets of Cantor space you get a Borel reduction to a part of the homeomorphism relation induced by a Polish group action (I'm not sure whether the entire homeomorphism relation is so induced).
So, in other words, any classification of open subsets of $\mathbb{R}^2$ up to homeomorphism is also a classification of any countable structure in any countable language up to isomorphism!  So, that's what I meant when I said it's "pretty darn hard."
So then what did "maybe not inconceivably hard" mean?  Some equivalence relations are so complicated that they are not classifiable by countable structures.  That is, they can not be Borel reduced to any isomorphism relation of $\mathcal{L}$-structures as described above.  For example, homeomorphism of closed subsets of Cantor space is classifiable by countable structures (Stone duality), while isomorphism of type $II_1$ von Neumann factors is not classifiable by countable structures (Sasyk-Törnquist 2009).  I have no idea where this problem sits with respect to this notion (to show that this homeomorphism relation is classifiable by countable structures it would be enough to show that there's a Borel way of associating with each open subset $U$ of the plane a closed subset of Cantor space homeomorphic to $U$'s space of ends).  For more examples see:
http://www.math.ucla.edu/~greg/223d.1.09f/countable.pdf
(*)Fine print disclaimer: not everybody thinks this is the "right" hierarchy to use.  One alternative is to "effectivize" everything (working lightface rather than boldface: think "recursive" instead of "Borel"), although this is quite challenging and restrictive.  But, aside from these, I'm not familiar with any notions with such a rich structure theory.  Note that I'm not including the "I'll know a classification when I see it" standpoint (which is fine for SCOTUS, but isn't particularly amenable to formal analysis).