[Someone]

This PDF only handles mapping N x N to N and vice versa, though.

For R x R, I would mix the bits of the two numbers

  [a0 a1 a2 a3...]
  [b0 b1 b2 b3...]

to

  [a0 b0 a1 b1 a2 b2 a3 b3...]

I am not sure that is a computable function for infinite-length bit sequences, though. Maybe none of those mapping functions are?

[sorokod]

Ah yes, I was thinking |N x N| = |N|. So is there a pairing function f: NxN -> N which can be used as compression?