Metacompact subspaces of products of ordinals

Abstract

Abstract. Let X be a subspace of the product of finitely many ordinals. X is countably metacompact, and X is metacompact iff X has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal. A theorem generalizing these two results is: X is λ-metacompact iff X has no closed subset homeomorphic to a (κ 1 , . . . , κn)-stationary set where κ 1 < λ. SourcesThis paper combines two lines of research. The first is the investigation of countably metacompact subspaces of the product of ordinals by Kemoto and Smith in [7] and [8]. A synthesis of the main theorems from these papers is proved that a subspace of a linearly ordered space is a D-space iff it is metacompact iff it has no closed subset homeomorphic to a stationary subset of a regular, uncountable cardinal. Stanley ([11] and [3]) proved the same equivalence for subspaces of the product of finitely many ordinals. Theorem 1.2. Let X be a subspace of the product of finitely many ordinals. The following are equivalent: X has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal.Kemoto, Tamano, and Yajima ([9]) proved that metacompactness, screenability, and weak submetalindelöfness are equivalent for subspaces of the product of two ordinals.Considering these results, it is natural to conjecture first, that every subspace of a finite product of ordinals is countably metacompact, and second, there is a