1. ## Designing a Function...

Hello!

I'm trying to construct a functions between 2 subsets of the 2d euclidean plane.

The sets are the Hawaiian earring. ie - the countable union of circles with centres (1/n,0) and radius 1/n,

and the countable union of circles of radius n & centres (n,0).

I can construct one using different n's, but is it possible to construct one using just x & y? eg - perhaps some sort of reciprocal function?

It seems like it should be the case, but I just can't see an equation! :-(

Any help would be greatly appreciated. :-)

Many thanks x

2. Originally Posted by TTB
I'm trying to construct a functions between 2 subsets of the 2d euclidean plane.

The sets are the Hawaiian earring. ie - the countable union of circles with centres (1/n,0) and radius 1/n,

and the countable union of circles of radius n & centres (n,0).

I can construct one using different n's, but is it possible to construct one using just x & y? eg - perhaps some sort of reciprocal function?
You can do this using the technique of geometric inversion. The map T defined by $\displaystyle T(x,y) = \Bigl(\tfrac{x}{x^2+y^2}, \tfrac{-y}{x^2+y^2}\Bigr)$ (inversion in the unit circle) takes circles through the origin to lines, and vice versa. It maps the Hawaiian earring to the family of vertical lines $\displaystyle x = n/2$, and it maps the other family of circles to the family of vertical lines $\displaystyle x=1/(2n)$.

Let S be the map defined by $\displaystyle S(x,y) = (1/(4x),y)$, which takes the first family of lines to the second family. Then the composite map TST will do what you want.