What is this space with infinitely many different points with distance $1$ between any two different points?

I'm reading Mac Lane's: Mathematics, Form and Function:

[...] There are also bizarre examples - such as "a space" with infinitely many different points, with distance $1$ between any two different points.

What is this bizarre space?

• There is not going to be any way to place this as a subset of anything familiar. You just need to take it on its own merits. – Will Jagy Mar 10 '14 at 3:20
• @WillJagy I don't get it. – Billy Rubina Mar 10 '14 at 3:38
• You can easily place a finite number of points in Euclidean space at mutual distance $1,$ but not infinitely many. As an alternative, we can take suitable multiples of $\cos n \theta$ and $\sin n \theta,$ where distance is given by the $L^2$ inner product on, say, continuous periodic functions on $[0,2 \pi].$ I guess that is the most concrete manifestation available. – Will Jagy Mar 10 '14 at 4:05
• Up to isometry, the quote you have is already a nearly complete definition of a space. The only thing it's missing is specifying which specific cardinality the space has. – Hurkyl Mar 10 '14 at 5:34

Define $d: M \times M \rightarrow M$ by
$$d(x, y) = 0 \iff x = y \;\;\text{and }$$ $$d(x, y) = 1 \iff x \neq y$$
This is what is called the discrete metric and is defined for any set $M$. You can show for yourself that $d$ actually constitutes a metric.
And the space $(M, d)$ is as you wish. The distance between any two different points is $1$. As long as you pick $M$ to be infinite, all your requirements are satisfied.
Take all the continuous functions that satisfy $f(x +2\pi) = f(x)$ for all real $x.$ Make a distance on these by $$d(f,g) = \sqrt{ \frac{1}{\pi} \int_0^{2 \pi} (f(x) - g(x))^2 dx }$$ With this distance function, we can make an infinite set of functions that are distance $1$ apart with $$f_n(x) = \frac{1}{\sqrt 2} \sin (nx).$$ The word "point" refers here to a function.