  Assuming Jensen's principle diamond, there is a compact Hausdorff space X
which is hereditarily Lindelof, hereditarily separable, and connected, such
that no closed subspace of X is both perfect and totally disconnected. The
Proper Forcing Axiom implies that there is no such space. The diamond example
also fails to satisfy the CSWP (the complex version of the Stone-Weierstrass
Theorem). This space cannot contain the two earlier examples of failure of the
CSWP, which were totally disconnected -- specifically, the Cantor set (W.
Rudin) and beta N (Hoffman and Singer).
