Selections and suborderability by G. Artico, U. Marconi, J.Pelant, L. Rotter, and M. Tkachenko

Selections and suborderability

G. Artico, U. Marconi, J.Pelant, L. Rotter, and M. Tkachenko

Research Announcement

A selection on an Hausdorff space X is a Vietoris continuous selection on all non-empty closed subsets of X. A weak selection is a continuous selection on all subsets consisting of one or two points. We extend results of J. van Mill and E. Wattel [Selections and orderability. Proc. Amer. Math. Soc. 83 (1981), no. 3, 601-605] about orderability of compact spaces.
A space is said to be suborderable if it is a subspace of a linearly ordered space. A suborderable space is hereditarily collectionwise normal.

Theorem 1 Let X be a completely regular space. The following are equivalent:

X² is pseudocompact and X has a weak selection.
X is suborderable and sequentially compact.
\beta X is orderable.

The equivalence of ii) and iii) is well-known.

Theorem 2 A pseudocompact space X with a weak selection is suborderable and sequentially compact provided it satisfies one of the following conditions:

X is a k-space.
X is countably (proto)compact.
X is first countable.
X is scattered.
X is connected.

Notice that a suborderable space is orderable when it is scattered or connected. Furthermore, a sequentially compact suborderable space is not necessarily orderable.

Theorem 3 Let G be a non-discrete locally pseudocompact topological group with a weak selection. Then G is locally compact, metrizable, and topologically orderable. In addition, G contains an open neighborhood of the identity which is homeomorphic either to the Cantor set or to the real line.

Corollary 4 A pseudocompact topological group with a weak selection is either finite or topologically homeomorphic to the Cantor set.

A zero-selection is a selection which associates an isolated point of F to every non-empty closed set F. We extend results of S. Fujii and T. Nogura [Characterizations of compact ordinal spaces via continuous selections. Topology Appl. 91 (1999), no. 1, 65-69] about well-orderability of compact spaces. Obviously a subspace of an ordinal space has a zero-selection.

Theorem 5 Let X be a space with a zero-selection. Then:

If X is locally compact and Lindelöf then it is homeomorphic to an ordinal space.
If X is locally compact and paracompact then it is embeddable into an open subspace of an ordinal space.
If X is pseudocompact then it is embeddable in an ordinal space; the image is open iff X is locally compact.

If X is scattered, its Cantor-Bendixson height is denoted by h(X).

Theorem 6 Let X be a locally compact locally countable space with the following properties:

X is hereditarily strongly collectionwise Hausdorff;
h(X) is a countable ordinal.

Then X is metrizable and homeomorphic to an open subspace of an ordinal space.

One may ask whether the condition on the height may be removed in the previous theorem. The answer is negative. Also paracompactness in claim 2 of Theorem 5 above cannot be easily weakened. This is shown in the following theorem.

Theorem 7 Assume the combinatorial property Diamond.

There exists a locally compact locally countable space which is hereditarily collectionwise normal and has a zero-selection, but it is not suborderable.
There exists a locally compact locally countable space with a zero-selection and Cantor-Bendixson height equal to 2 which is not normal.

These results have been presented in Trieste and Warsaw (May 2000) and La Manga, Murcia (June 2000). The preprint with these results will be submitted soon for its publication in a journal.

E-mail: umarconi@math.unipd.it, pelant@beba.cesnet.cz, mich@xanum.uam.mx.

Date Received: July 7, 2000.