Topology Atlas Document # iaai-18 | © 2000 Copyright by Umberto Marconi.

Some questions about selections and orderability

Umberto Marconi

All spaces are assumed to be completely regular.
A topological space X is said to be weakly orderable if there exists a linear order on X such that the open interval topology is weaker than the original one. Obviously a weakly orderable compact space is orderable.

A weak selection on X is a continuous selection on all subsets consisting of two points.

It is well-known that if X is connected (Michael, 1951, [10]) or compact (van Mill-Wattel, 1981, [12]) then the existence of a weak selection implies that X is weakly orderable.

J. van Mill and E. Wattel put the following question [12]:

Question 1. Is X weakly orderable whenever X admits a weak selection?
There are some partial answers to Question 1 (e.g., see [3]), but a decisive answer has not yet been provided. I don't know if there exists an answer by assuming the stronger hypothesis that there exists a continuous selection on all non-empty closed subsets of X.

E. K. van Douwen proved that a countably compact space with a weak selection is sequentially compact [4].

Question 2. Let X be a pseudocompact space with a weak selection. Is it true that X is sequentially compact?
In view of Theorem 1 of [3], Question 2 is equivalent to the following:
Question 2'. Let X be a pseudocompact space with a weak selection. Is it true that X2 is pseudocompact?

References

  1. G. Artico and U. Marconi and R. Moresco and J. Pelant, Selectors and scattered spaces, Topol. Appl., Proc. Erice Course on Convergence and Topology, to appear (2000), 14 pages.
  2. G. Artico and U. Marconi, Selections and topologically well-ordered spaces, Topol. Appl., to appear (2000), 6 pages.
  3. G. Artico, U. Marconi, J.Pelant, L. Rotter, and M. Tkachenko, Selections and suborderability, Topology Atlas Reseaarch Announcement (2000). http://at.yorku.ca/i/a/a/i/09.htm
  4. E.K. van Douwen, Mappings from hyperspaces and convergent sequences, Topol. Appl. 34 (1990), 35-45.
  5. S. Eilenberg, Ordered topological spaces, Amer. J. Math. 63 (1941), 39-45.
  6. R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Inventiones Math., 6 (1968), 150-158.
  7. S. Fujii and T. Nogura, Characterizations of compact ordinal spaces via continuous selections, Topol. Appl. 20 (1997), 1-5.
  8. H. Herrlich, Ordnungsfähigkeit total-diskontinuierliche Räume, Math. Ann. 159 (1965), 77-80.
  9. D. Kurepa, Sur l'écart abstrait, Glasnik Mat. Fiz. Astr. 11 (1956), 105-134.
  10. E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.
  11. J. van Mill, J. Pelant, and R. Pol, Selections that characterize topological completeness, Fund. Math. 149 (1996), 127-141.
  12. J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), no. 3, 601-605.
  13. J. van Mill and E. Wattel, Orderability from selections: Another solution to the orderability problem, Fund. Math. 121 (1984), 219-229.
  14. S. Purisch, On the orderability of Stone-Cech compactifications, Proc. Amer. Math. Soc. 41 (1973), 55-56.
  15. S. Purisch, Scattered compactifications and the orderability of scattered spaces, II, Proc. Amer. Math. Soc. 95 (1985), 636-640.
  16. S. Purisch, A History of results on orderability and suborderability, Topology Atlas Invited Contributions 1 (1996). "http://at.yorku.ca/t/a/i/c/18.htm
Date Received: September 27, 2000.
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