In mathematics, an η set (eta set) is a type of totally ordered set introduced by Hausdorff (1907, p. 126, 1914, chapter 6 section 8) that generalizes the order type η of the rational numbers.
The only non-empty countable η0 set (up to isomorphism) is the ordered set of rational numbers.
Suppose that κ = ℵα is a regular cardinal and let X be the set of all functions f from κ to {−1,0,1} such that if f(α) = 0 then f(β) = 0 for all β > α, ordered lexicographically.
The union of all these sets is the class of surreal numbers.
A dense totally ordered set without endpoints is an ηα set if and only if it is ℵα saturated.
For any given ordinal α, any two ηα sets of cardinality ℵα are isomorphic (as ordered sets).