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Orderability in the presence of local compactness. (English) Zbl 1148.54014

All spaces considered are at least Hausdorff. The article is motivated by results discussed in [J. van Mill and E. Wattel, Proc. Am. Math. Soc. 83, 601-605 (1981; Zbl 0473.54010)].
It is shown that a locally compact paracompact space has a continuous selection for its Vietoris hyperspace of nonempty closed subsets if and only if it is a topologically well-orderable subspace of some orderable space.
The author also proves that a locally compact paracompact space is suborderable if and only if it has a continuous weak selection. His investigations show that every locally compact paracompact space which has a continuous weak selection is semi-orderable, that is, is the topological sum of two orderable spaces.

MSC:

54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54B20 Hyperspaces in general topology
54C65 Selections in general topology

Citations:

Zbl 0473.54010
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References:

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