[dwohnitmok]

That's fair. "Squishing the integers" was a poor choice of words. "Squishing" an undistinguished countable set is perhaps a better analogy.

In particular I was thinking of undistinguished countable sets (although I was confusingly throwing in ordering of the integers to try to make my point more accessible) that you then add topological structure on top of.

In that world the integers are simply the discrete topology on a countable set. Or more explicitly (to contrast with the next definition), where all singleton sets are open.

The rationals then are formed by any metric whose induced topology does not include singleton sets.

That is, any attempt to uniformly bring elements "closer" than the world where single points are open gives rise to a topology homeomorphic to the rational numbers.

[SamReidHughes]

The thing that surprised me yesterday (since I don't know much about topology) is that every metrizable countable set without isolated points is the same.

So the numbers k/2^n, the points on the unit circle with rational y/x, or the set Q \ Z, are homeomorphic.

The fact that you can slice and dice the rationals is counterintuitive.

[SamReidHughes]

Update: it's not counterintuitive any more.