Lexicographic exponentiation of chains

Abstract

The lexicographic power Δ ^Γ of chains Δ and Γ is, roughly, the Cartesian power ∏_{γ ∈ Γ}Δ, totally ordered lexicographically from the left. Here the focus is on certain powers in which either Δ = ℝ or Γ = ℝ, with emphasis on when two such powers are isomorphic and on when Δ ^Γ is 2-homogeneous. The main results are:

(1) For a countably infinite ordinal α, ℝ_{α ^* +α}≃ℝ ^α.

(2) ℝ ^ℝ≂ℝ ^ℚ.

(3) For Δ a countable ordinal ≥ 2, Δ ^ℝ, with its smallest element deleted, is 2-homogeneous.