---
abstract: 'We consider the *Complex Stone-Weierstrass Property* (CSWP), which is the complex version of the Stone-Weierstrass Theorem. If $X$ is a compact subspace of a product of three linearly ordered spaces, then $X$ has the CSWP if and only if $X$ has no subspace homeomorphic to the Cantor set. In addition, every finite power of the double arrow space has the CSWP. These results are proved using some results about those compact Hausdorff spaces which have scattered-to-one maps onto compact metric spaces.'
author:
- 'Kenneth Kunen[^1] [^2]'
title: 'Ordered Spaces, Metric Preimages, and Function Algebras [^3] '
---

Introduction {#sec-intro}
============

All topologies discussed in this paper are assumed to be Hausdorff. As usual, a subset of a space is *perfect* iff it is closed and non-empty and has no isolated points, so $X$ is *scattered* iff $X$ has no perfect subsets.

The usual version of the *Stone-Weierstrass Theorem* involves subalgebras of $C(X,{{\mathbb R}})$, and is true for all compact $X$. If one replaces the real numbers ${{\mathbb R}}$ by the complex numbers ${{\mathbb C}}$, the “theorem” is true for some $X$ and false for others, so it becomes a *property* of $X$:

\[def-cswp\] If $X$ is compact, then 