ITES2001 Fourth Italian-Spanish conference on General Topology and its applications

Bressanone, 27-30 June 2001

Selectors and topological properties of spaces

Jan Pelant

Mathematical Institute of Academy of Sciences, Czech Republic

We study continuous selectors σ : 𝔉_{τ}(X) → X where 𝔉_{τ}(X) is a hyperspace of non-empty closed subsets of X equipped with a topology τ and σ(F) ∈ F for each F ∈ 𝔉(X). This topic has been investigated for years and we are going to mention some new results and to relate them to other results in the literature.

Some sample facts from our background: Theorem 1 (Mazurkiewicz-Sierpiński). X is first countable scattered compact

⟺ X is homeomorphic to a countable ordinal ⟺ X is compact and countable. A zero-selector on X is a selector σ such that σ(F) is isolated relatively to F. If X has a zero-selector, then X is scattered. Moreover, any subspace of ordinals

has a zero-selector–just take minima of non-empty closed sets.

A first-countable paracompact scattered space is a completely metrizable subspace of an ordinal space (Telgársky); consequently it has a zero-selector.

Theorem 2 (Fujii-Nogura). If X is compact and there exist a zero-selector, then X is homeomorphic to an ordinal space.

New and also older results mentioned in this lecture were obtained jointly with G.Artico, U.Marconi, L.Rotter and M.Tkachenko.

References

[1] G. Artico and U. Marconi, Selections and topologically well-ordered spaces, Topology Appl., to appear (2000), 6 pages.

[2] G. Artico, U. Marconi, J. Pelant, L. Rotter, M. Tkachenko,Selections and suborderability, submitted.

[3] V. Gutev and T. Nogura, Fell continuous selections and topologically well-orderable spaces, Preprint (1999), 8 pages.

[4] S. Fujii and T. Nogura, Characterizations of compact ordinal spaces via continuous selections, Topology Appl., 20 (1997), 1–5.

[5] J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), 601–605.

[6] J. van Mill and E. Wattel, Orderability from selections: Another solution to the orderability problem, Fund. Math. 121 (1984), 219–229.

[7] S. Purish, Scattered compactifications and the orderability of scattered spaces, II, Proc. Amer. Math. Soc. 95 (1985), 636–640.

[8] R. Telgársky, Total paracompactness and paracompact dispersed spaces, Bull. Polish Acad. Sci. Math. 16 (1968), 567–572.