Vietoris continuous selections and disconnectedness-like properties
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- by Valentin Gutev and Tsugunori Nogura PDF
- Proc. Amer. Math. Soc. 129 (2001), 2809-2815 Request permission
Abstract:
Suppose that $X$ is a Hausdorff space such that its Vietoris hyperspace $({\mathcal {F}}(X),\tau _{V})$ has a continuous selection. Do disconnectedness-like properties of $X$ depend on the variety of continuous selections for $({\mathcal {F}}(X),\tau _{V})$ and vice versa? In general, the answer is “yes” and, in some particular situations, we were also able to set proper characterizations.References
- G. Artico, U. Marconi, R. Moresco and J. Pelant, Selectors and Scattered Spaces, Topology Appl., to appear.
- Daniela Bertacchi and Camillo Costantini, Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, Topology Appl. 88 (1998), no. 3, 179–197. MR 1632069, DOI 10.1016/S0166-8641(97)00175-2
- M. Choban, Many-valued mappings and Borel sets. I, Trans. Moscow Math. Soc. 22 (1970), 258–280.
- C. Costantini and V. Gutev, Recognizing special metrics by topological properties of the “metric”-Proximal hyperspace, preprint.
- R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150–158. MR 244959, DOI 10.1007/BF01425452
- V. Gutev and T. Nogura, Selections for Vietoris-like hyperspace topologies, Proc. London Math. Soc. 80 (1) (2000), 235–256.
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Jan van Mill and Evert Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), no. 3, 601–605. MR 627702, DOI 10.1090/S0002-9939-1981-0627702-4
- Tsugunori Nogura and Dmitri Shakhmatov, Characterizations of intervals via continuous selections, Rend. Circ. Mat. Palermo (2) 46 (1997), no. 2, 317–328. MR 1617361, DOI 10.1007/BF02977032
- Tsugunori Nogura and Dmitri Shakhmatov, Spaces which have finitely many continuous selections, Boll. Un. Mat. Ital. A (7) 11 (1997), no. 3, 723–729 (English, with Italian summary). MR 1489043
- Teodor C. Przymusiński, On the dimension of product spaces and an example of M. Wage, Proc. Amer. Math. Soc. 76 (1979), no. 2, 315–321. MR 537097, DOI 10.1090/S0002-9939-1979-0537097-3
- M. L. Wage, The dimension of product spaces, preprint (1977).
- Michael L. Wage, The dimension of product spaces, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 10, 4671–4672. MR 507930, DOI 10.1073/pnas.75.10.4671
Additional Information
- Valentin Gutev
- Affiliation: School of Mathematical and Statistical Sciences, Faculty of Science, University of Natal, King George V Avenue, Durban 4041, South Africa
- Email: gutev@sci.und.ac.za
- Tsugunori Nogura
- Affiliation: Department of Mathematics, Faculty of Science, Ehime University, Matsuyama, 790 Japan
- Email: nogura@ehimegw.dpc.ehime-u.ac.jp
- Received by editor(s): November 17, 1999
- Received by editor(s) in revised form: January 17, 2000
- Published electronically: February 9, 2001
- Communicated by: Alan Dow
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2809-2815
- MSC (2000): Primary 54C65, 54B20, 54F45
- DOI: https://doi.org/10.1090/S0002-9939-01-05883-X
- MathSciNet review: 1838807