Η set
# Η set

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.

## 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 $η_0$ 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 an $η_α$ 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 $Σ_{β<α}$ $2^{ℵβ}$ ≤ $ℵ_α.$

