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Transactions of the American Mathematical Society

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The orderability and closed images of scattered spaces


Author: S. Purisch
Journal: Trans. Amer. Math. Soc. 320 (1990), 713-725
MSC: Primary 54F05; Secondary 54D99
DOI: https://doi.org/10.1090/S0002-9947-1990-0989584-0
MathSciNet review: 989584
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Abstract: A (totally) orderable scattered space and a space homeomorphic to a subspace of an ordinal space are characterized in terms of a neighborhood subbase for each of their points plus what corresponds to a neighborhood base for each of their non-$ Q$-gaps. These generalize the characterizations in [ P$ _{1}$] of an orderable compact scattered space and in [B] of a space homeomorphic to a compact ordinal space. Generalizing a result in [M] it is shown that a space is orderable and scattered iff it is the $ 2$ to $ 1$ image under a closed map of a subspace of an ordinal space. In response to a question of Telgarsky [T] a simple description is given of a closed map with discrete fibers from an orderable scattered space onto an orderable perfect space. Maps that preserve length conditions on a scattered space are touched upon.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0989584-0
Keywords: Totally orderable, suborderable, scattered, strongly collectionwise Hausdorff, ordering property, ordinal property, stationary set, interlacing property, sequence, length, $ Q$-gap, greatest ordered compactification, monotone normality, far point
Article copyright: © Copyright 1990 American Mathematical Society