η set

In mathematics, an η 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.

Definition

If α is an ordinal then a η_α set is a totally ordered set such that if X and Y are two subsets of cardinality less than ℵ_α such that 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=ℵ_α 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 η_α set. The union of all these sets is the class of surreal numbers.

A dense totally ordered set without endpoints is a η_α set if and only if it is ℵ_α saturated.

Properties

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

For any given ordinal α, any two η_α sets of cardinality ℵ_α are isomorphic (as ordered sets). An η_α set of cardinality ℵ_α exists if ℵ_α is regular and ∑_β<α 2^ℵ_β ≤ ℵ_α.

References

• English translation in Hausdorff (2005)