3 ms·

You can get from R^d to I^d (with I the open unit interval) easily by applying x -> (arctan(x) + pi/2)/pi and y -> tan(pi * y - pi/2) to each coordinate. There

by Skeime 3y ago

You can get from R^d to I^d (with I the open unit interval) easily by applying x -> (arctan(x) + pi/2)/pi and y -> tan(pi * y - pi/2) to each coordinate. There are also ways to go from R to the closed unit interval. You can then combine this with space-filling curves for I^d.

yarg 3y ago
You can even go from R to I^d. The problem that I'm having is in getting from R to R^d, which I still don't think is possible - look at the comment where I mention "surjections" and "bijections".
- Skeime 3y ago
  By applying the map from I^d to R^d (using the tan-arctan-construction above in each coordinate) after the map from R to R^d, you get a map from R to R^d.
  - yarg 3y ago
    I^n is just a countably infinite subset of R^n.
    - Skeime 3y ago
      No, I^n is also uncountable. In fact, the usual proof for the uncountability of R is really proof of the uncountability of I, and then the natural injection into R is used to extend it to R.