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Ordered spaces with special bases

Harold Bennett, David Lutzer (1998)

Fundamenta Mathematicae

We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a G δ -diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered...

Order-like structure of monotonically normal spaces

Scott W. Williams, Hao Xuan Zhou (1998)

Commentationes Mathematicae Universitatis Carolinae

For a compact monotonically normal space X we prove:   (1)   X has a dense set of points with a well-ordered neighborhood base (and so X is co-absolute with a compact orderable space);   (2)   each point of X has a well-ordered neighborhood π -base (answering a question of Arhangel’skii);   (3)   X is hereditarily paracompact iff X has countable tightness. In the process we introduce weak-tightness, a notion key to the results above and yielding some cardinal function results on monotonically normal...

Ordinal remainders of classical ψ-spaces

Alan Dow, Jerry E. Vaughan (2012)

Fundamenta Mathematicae

Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain T α : α < λ of infinite subsets of ω, there exists ℳ ⊂ [ ω ] ω , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain T α : α < λ , hence a ψ-space with Stone-Čech remainder...

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