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--gaps. These generalize the characterizations in [ P] 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 to 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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Additional Information

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,
-gap,
greatest ordered compactification,
monotone normality,
far point

Article copyright:
© Copyright 1990
American Mathematical Society