Useful to map an infinite set onto a finite space?

Question

Take an infinite set such as the integers, rationals or primes and say you were able to map the elements of the set into a finite space. For example maybe you could map each element of the integers or primes onto a line or loop of finite length. In order to do this my guess is that you would need the elements of the set to really converge rapidly such that if you took the limit of the set tending towards infinity you get a finite result. Would this be a useful thing to do? How so? Obviously you can't literally map each element of an infinite set but you can always map the next one. Also just curious, what does the mapping $ \Bbb Z \to \zeta(\Bbb Z) $ look like/is it useful?

Answer 1

There are two different concepts.

One is finiteness and the other is boundedness.

For example the unit interval $(0,1)$ is infinite and bounded. On the other hand the set of natural numbers is infinite and unbounded.

Now if you look at the reciprocals of natural numbers, $ \{ 1,1/2, 1/3, 1/4, \}$ you get an infinite set which is bounded.

Every finite set is bounded but infinite sets could be bounded or unbounded. We can transform an infinite set into a bounded set with some transformations, such as the example $f(n)= 1/n$ for natural numbers.

Answer 2

There are many mappings that take the reals into a finite interval. Some examples are the inverse tangent and hyperbolic tangent functions. The reals, of course, include integers, rationals, primes and so they would all get mapped into the open interval $\,(-1,1).$