Compact spaces of diversity two
The diversity of a space is the number of homeomorphism types of its open subsets; for example the Cantor set has diversity 2 but a converging sequence has diversity \(\aleph_0\). It was shown by \textit{J. Mioduszewski} [Colloq. Math. 39, 35-40 (1978; Zbl 0401.54007)] that a compact space \(X\) of diversity 2 is homogeneous, dense-in-itself and zero-dimensional, and that each noncompact open subset of \(X\) is homeomorphic to \(\omega \times X\); this implies that such spaces are hereditarily Lindelöf. The authors concentrate on compact ordered spaces and on products of diversity 2. All compact ordered spaces of diversity 2 are seen to be obtained by splitting the points of a dense subset of either the unit interval or a connected Suslin line. This gives one a structural handle on such spaces and allows the authors to describe a variety of such spaces and characterize some of them. As to products: the authors prove that the product of spaces of diversity 2 is again of diversity 2, provided it is hereditarily Lindelöf; as a corollary it follows that only the Cantor set has a square of diversity 2. From the conjunction of \(\text{MA} (\aleph_1)\) and ``all \(\aleph_1\)-dense subsets of \(\mathbb{R}\) are isomorphic'' deduce ``if \(X\) and \(Y\) are compact, ordered and of diversity 2 and if \(X\times Y\) is also of diversity 2 then \(X\) or \(Y\) is the Cantor set''. By contrast, from \(\diamondsuit\) the authors construct two Suslin lines of diversity 2 whose product also has diversity 2.
scientific article; zbMATH DE number 896602
scientific article