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Every collectionwise Hausdorff (CWT2) first countable space is regular. Generalizations of first countability are considered. If X is hereditarily CWT2(HCWT2), blobular, of character N1, and contains no first countable S space, then X is regular. The existence of a first countable S space implies the existence of a nonregular, HCWT2, lob space of character N1. If it is consistent that there are no...
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...
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