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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The orderability and suborderability of metrizable spaces


Author: S. Purisch
Journal: Trans. Amer. Math. Soc. 226 (1977), 59-76
MSC: Primary 54F05; Secondary 54E35, 06A05
MathSciNet review: 0428296
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Abstract: A space is defined to be suborderable if it is embeddable in a (totally) orderable space. It is shown that a metrizable space X is suborderable iff (1) each component of X is orderable, (2) the set of cut points of each component of X is open, and (3) each closed subset of X which is a union of components has a base of clopen neighborhoods. Note that condition (1) and hence this result is topological since there are many good topological characterizations of connected orderable spaces.

In a space X let Q denote the union of all nondegenerate components each of whose noncut points has no compact neighborhood. It is also shown that a metrizable space X is orderable iff (1) X is suborderable, (2) $ X - Q$ is not a proper compact open subset of X, and (3) if W is a neighborhood of $ p \in X$ and K is the component in X containing p such that $ (W - K) - Q$ has compact closure and $ \{ p\} $ is the intersection of the closures of $ (W - K) - Q$ and $ (W - K) \cap Q$, then K is a singleton. Corollaries are given; every condition in each of these corollaries is concisely stated and sufficient for a space to be orderable when it is metrizable and suborderable.

Both of these results are extended to a class properly containing the metrizable spaces.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0428296-2
PII: S 0002-9947(1977)0428296-2
Keywords: Totally orderable space, suborderable space, metrizable space, convex metric, cut point, clopen neighborhood base, $ \varepsilon $-decomposition, large inductive dimension zero, neighbors, ordered compactification
Article copyright: © Copyright 1977 American Mathematical Society