Complex function algebras and removable spaces

The compact Hausdorff space X has the Complex Stone–Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone–Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X has the CSWP if and only if X does not contain a copy of the Cantor set.