Η set

In mathematics, an η set (eta set) is a type of totally ordered set introduced by that generalizes the order type η of the rational numbers.

Definition
If $$\alpha$$ is an ordinal then an $$\eta_\alpha$$ set is a totally ordered set in which for any two subsets $$X$$ and $$Y$$ of cardinality less than $$\aleph_\alpha$$, if every element of $$X$$ is less than every element of $$Y$$ then there is some element greater than all elements of $$X$$ and less than all elements of $$Y$$.

Examples
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. Then X is a ηα set. 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.

Properties
Any ηα set X is universal for totally ordered sets of cardinality at most ℵα, meaning that any such set can be embedded into X.

For any given ordinal α, any two ηα sets of cardinality ℵα are isomorphic (as ordered sets). An ηα set of cardinality ℵα exists if ℵα is regular and Σβ<α 2ℵβ ≤ ℵα.