Reals have “gaps”, too. https://en.wikipedia.org/wiki/Surreal_number#Overview:

”There are also representations like

{ 0, 1, 2, 3, … | } = ω

{ 0 | 1, 1/2, 1/4, 1/8, … } = ε

where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number.”

So, ε is a gap just above zero, π-ε a gap just below π, etc.

(The difference is that, when stepping from real to real, you won’t accidentally step on a surreal in the way you’ll frequently hit irrationals when you step from rational to rational)