What Sets Exhibit the Wada Property in R2, R3, and R?

[tt2348]

What kind of sets exhibit the wada property? I know R2 does but does it extend to R3 or R itself?

 

[quasar987]

The lakes (or basins) of Wada are an example in the plane of 3 disjoint sets sharing the same boundary. You get an analogous (n+2)-dimensional construction by "elevating" in the additional dimensions. Namely, if B,C,D are the 3 disjoint sets in R², take B x R^n, C x R^n, D x R^n, where x is the cartesian product.

This takes care of every dimension except the first. In R, an example of 3 disjoint sets sharing the same boundary is A = {the points of the form 1/2n, n positive integer}, B= {the points of the form 1/(2n+1), n positive integer}, C={the points of the form -1/n, n positive integer}. Their common boundary being 0.

 

[tt2348]

Don't the sets have to be open? Yes they do, also -1/n also has a boundary at -1, and 1/2n+1 and 1/2n have boundary points at 1/2, 1/3, so they don't have the same boundary.

 

[quasar987]

Oh yes, you are right on all accounts. Plus, the sets have to be connected. So my exemple in R isn't good.

Connected open sets in R are open intervals. Clearly, no three open intervals can have a common boundary: call I_1 the interval containing -\infty and I_2 the interval containing +\infty. Then if I_1=(-\infty,a) and I_2=(b,
+\infty), it must be that a=b if these two are to have the same boundary. But then I_1 u I_2 = R\{a} and so there is no place to put the third interval I_3.

I wonder if it works if we remove the connectivity condition.

 

[tt2348]

I was trying to think of a topology that maybe the open sets have odd properties on R. I was told that maybe the complement of the cantor set would work , but I think that may be wrong too

 

[tt2348]

https://www.physicsforums.com/showthread.php?t=380692

that was a problem I was working on, I assumed that each set was composed of connected components
any input you have would be appreciated

 

[quasar987]

Well, what odd properties are you after?

 

[tt2348]

Well I'm not sure.. Odd properties that make this problem easier? But it's kind of impossible to change the fact that Any open set will have an interval