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Compact spaces of diversity two. (English) Zbl 0854.54023
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 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.
Reviewer: K.P.Hart (Delft)

##### MSC:
 54D30 Compactness 54A35 Consistency and independence results in general topology 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54B10 Product spaces in general topology 06A05 Total orders 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 03E50 Continuum hypothesis and Martin’s axiom 03E65 Other set-theoretic hypotheses and axioms 03E05 Other combinatorial set theory 03C25 Model-theoretic forcing
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