Compact spaces of diversity two

A compact infinite Hausdorff space X is diversity two if all nonempty clopen subsets are homeomorphic to X and all noncompact open subsets are homeomorphic to each other. We study and attempt to classify such spaces. A sufficient basis condition is found for a compact space to be diversity two. A characterization is found for compact diversity two, (totally) orderable, separable spaces which have no uncountable metrizable subspaces. Compact Souslin lines of diversity two are constructed and studied. For compact diversity two orderable spaces whose product is diversity two, it is shown that it is independent of ZFC whether one of the factors must be the Cantor set. The product of compact diversity two spaces will have diversity two iff it is hereditarily Lindelِf. Results on diversity two products are obtained via some new results on hereditarily Lindelِf products of more general spaces. An interesting result proved along the way is: the set of nonjump points is scattered in a compact, first countable, ordered space iff under some admissible order on the space each point is a jump point.