A strangely connected subset of $\Bbb R^2$ Let $S\subset{\Bbb R}^2$ (or any metric space, but we'll stick with $\Bbb R^2$) and let $x\in S$.  Suppose that all sufficiently small circles centered at $x$ intersect $S$ at exactly $n$ points; if this is the case then say that the valence of $x$ is $n$.  For example, if $S=[0,1]\times\{0\}$, every point of $S$ has valence 2, except $\langle0,0\rangle$ and $\langle1,0\rangle$, which have valence 1.
This is a typical pattern, where there is an uncountable number of 2-valent points and a finite, possibly empty set of points with other valences. In another typical pattern, for example ${\Bbb Z}^2$, every point is 0-valent; in another, for example a disc, none of the points has a well-defined valence.
Is there a nonempty subset of $\Bbb R^2$ in which every point is 3-valent? I think yes, one could be constructed using a typical transfinite induction argument, although I have not worked out the details. But what I really want is  an example of such a set that can be exhibited concretely.
What is it about $\Bbb R^2$ that everywhere 2-valent sets are well-behaved, but 
everywhere 3-valent sets are crazy? Is there some space we could use instead of $\Bbb R^2$ in which the opposite would be true?

Here are the details of the transfinite induction argument:
Well-order the set of points in $\mathbb{R}^d$ and let $p_\alpha$ denote the point at ordinal index $\alpha$.  Then define:
$S_{<\alpha} = \bigcup_{\beta \lt \alpha}{S_\beta}$
$S_0 = \{p_0\}$
$S_\alpha = S_{<\alpha}$ when there is a $(d-1)$-sphere centered at $p_\alpha$ intersecting $S_{<\alpha}$ at more than $n$ points.
$S_\alpha = S_{<\alpha} \cup \{p_\alpha\}$ otherwise.
It can be seen that $\bigcup_{\alpha}{S_\alpha}$ is everywhere-$n$-valent.
This is actually a little bit stronger since every point has $n$ neighbors at every distance, not just arbitrarily small ones.  It works for any $n$ and also any $d$, but I don't know exactly which metric spaces.
I don't know if there is any concrete example.  Maybe it is worth considering the valency of plane fractals like the Koch snowflake; perhaps there are concrete examples of everywhere-$2 \cdot n$-valent curves that are wiggly enough to enter and exit arbitrarily small circles multiple times.  Because of the Jordan curve theorem, this approach seems less promising for finding everywhere-odd-valency sets.
