×

Consonance and Cantor set-selectors. (English) Zbl 1262.54004

For a topological space \(X\), let \({\mathcal F}(X)\) denote the collection of all nonempty closed subsets of \(X\). A space is said to be consonant if the co-compact topology on \({\mathcal F}(X)\) coincides with the upper Kuratowski topology. A mapping \(\varphi : Z \to {\mathcal F}(X)\) is said to be lower semicontinuous (respectively, upper semicontinuous) if for every open (respectively, closed) subset \(A\) of \(X\), the set \(\varphi ^{-1} [A] =\{ z \in Z : \varphi(z) \cap A \neq \varnothing\}\) is open (respectively, closed) in \(Z\). A subset \(A \subset X\) is called a section for \(\varphi :Z \to {\mathcal F}(X)\) if \(A \cap \varphi (z) \neq \varnothing\) for every \(z \in Z\). A. Bouziad [Proc.Am.Math.Soc.127, No.12, 3733–3737 (1999; Zbl 0976.54036)] proved that a space \(X\) is consonant if and only if for every (not necessarily Hausdorff) compact space \(Z\) and for every lower semicontinuous mapping \(\varphi : Z \to {\mathcal F}(X)\), every open section \(U \subset X\) for \(\varphi\) contains a compact section \(K\subset X\) for \(\varphi\).
On the other hand, let \(\mathfrak{C}\) denote the Cantor space. A space \(X\) is called a \(\mathfrak{C}\)-selector (or Cantor set-selector) if for every lower semicontinuous mapping \(\varphi : \mathfrak{C} \to {\mathcal F}(X)\), there exists an upper semicontinuous mapping \(\theta : \mathfrak{C} \to {\mathcal F}(X)\) such that \(\theta (z) \subset \varphi (z)\) for every \(z \in \mathfrak{C}\).
In this paper, the author proves that every metrizable consonant space is a \(\mathfrak{C}\)-selector, which gives an affirmative answer to a question posed by him in [Set-Valued Anal.15, No. 3, 275–295 (2007; Zbl 1140.54003)]. The author also proves that a metrizable space \(X\) is consonant if and only if the space of all compact subsets of \(X\) with the Vietoris topology is consonant.

MSC:

54C60 Set-valued maps in general topology
54C65 Selections in general topology
54B20 Hyperspaces in general topology
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Alleche B., Calbrix J., On the coincidence of the upper Kuratowski topology with the cocompact topology, Topology Appl., 1999, 93(3), 207-218 http://dx.doi.org/10.1016/S0166-8641(97)00269-1; · Zbl 0943.54013
[2] Banakh T.O., Topology of spaces of probability measures. I. The functors P τ and P̃, Mat. Stud., 1995, 5, 65-87 (in Russian); · Zbl 1023.28501
[3] Borges C.J.R., A study of multivalued functions, Pacific J. Math., 1967, 23, 451-461; · Zbl 0153.24204
[4] Bouziad A., Borel measures in consonant spaces, Topology Appl., 1996, 70(2-3), 125-132 http://dx.doi.org/10.1016/0166-8641(95)00089-5; · Zbl 0852.54009
[5] Bouziad A., A note on consonance of P δ subsets, Topology Appl., 1998, 87(1), 53-61 http://dx.doi.org/10.1016/S0166-8641(97)00131-4; · Zbl 0946.54009
[6] Bouziad A., Consonance and topological completeness in analytic spaces, Proc. Amer. Math. Soc., 1999, 127(12), 3733-3737 http://dx.doi.org/10.1090/S0002-9939-99-04902-3; · Zbl 0976.54036
[7] Bouziad A., Filters, consonance and hereditary Baireness, Topology Appl., 2000, 104(1-3), 27-38 http://dx.doi.org/10.1016/S0166-8641(99)00014-0;
[8] Choban M.M., Many-valued mappings and Borel sets. I, Trans. Moscow Math. Soc., 1970, 22, 258-280; · Zbl 0236.54012
[9] Costantini C., Watson S., On the dissonance of some metrizable spaces, Topology Appl., 1998, 84(1-3), 259-268 http://dx.doi.org/10.1016/S0166-8641(97)00096-5; · Zbl 0966.54005
[10] Debs G., Espaces héréditairement de Baire, Fund. Math., 1988, 129(3), 199-206; · Zbl 0656.54023
[11] Dolecki S., Greco G.H., Lechicki A., Sur la topologie de la convergence supérieure de Kuratowski, C. R. Acad. Sci. Paris, 1991, 312(12), 923-926; · Zbl 0789.54009
[12] Dolecki S., Greco G.H., Lechicki A., When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?, Trans. Amer. Math. Soc., 1995, 347(8), 2869-2884 http://dx.doi.org/10.1090/S0002-9947-1995-1303118-7; · Zbl 0845.54005
[13] El’kin A.G., A-sets in complete metric spaces, Dokl. Akad. Nauk SSSR, 1967, 175, 517-520;
[14] Gutev V., Selections and approximations in finite-dimensional spaces, Topology Appl., 2005, 146-147, 353-383 http://dx.doi.org/10.1016/j.topol.2003.06.002; · Zbl 1066.54022
[15] Gutev V., Completeness, sections and selections, Set-Valued Anal., 2007, 15(3), 275-295 http://dx.doi.org/10.1007/s11228-007-0041-0; · Zbl 1140.54003
[16] Gutev V., Nedev S., Pelant J., Valov V., Cantor set selectors, Topology Appl., 1992, 44(1-3), 163-166 http://dx.doi.org/10.1016/0166-8641(92)90089-I; · Zbl 0769.54020
[17] Gutev V., Valov V., Sections, selections and Prohorov’s theorem, J. Math. Anal. Appl., 2009, 360(2), 377-379 http://dx.doi.org/10.1016/j.jmaa.2009.06.063; · Zbl 1185.28022
[18] Koumoullis G., Cantor sets in Prohorov spaces, Fund. Math., 1984, 124(2), 155-161; · Zbl 0562.54052
[19] Kuratowski K., Topology. I, Academic Press, New York-London; PWN, Warsaw, 1966;
[20] Michael E., Continuous selections. I, Ann. of Math., 1956, 63, 361-382 http://dx.doi.org/10.2307/1969615; · Zbl 0071.15902
[21] Michael E., Continuous selections. II, Ann. of Math., 1956, 64, 562-580 http://dx.doi.org/10.2307/1969603; · Zbl 0073.17702
[22] Michael E., A theorem on semi-continuous set-valued functions, Duke Math. J., 1959, 26, 647-651 http://dx.doi.org/10.1215/S0012-7094-59-02662-6; · Zbl 0151.30805
[23] van Mill J., Pelant J., Pol R., Selections that characterize topological completeness, Fund. Math., 1996, 149(2), 127-141; · Zbl 0861.54016
[24] Nedev S.J., Valov V.M., On metrizability of selectors, C. R. Acad. Bulgare Sci., 1983, 36(11), 1363-1366; · Zbl 0549.54013
[25] Nogura T., Shakhmatov D., When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies?, Topology Appl., 1996, 70(2-3), 213-243 http://dx.doi.org/10.1016/0166-8641(95)00098-4; · Zbl 0848.54007
[26] Preiss D., Metric spaces in which Prohorov’s theorem is not valid, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1973, 27, 109-116 http://dx.doi.org/10.1007/BF00536621; · Zbl 0255.60002
[27] Prokhorov Yu.V., Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl., 1956, 1(2), 157-214 http://dx.doi.org/10.1137/1101016;
[28] Przymusinski T., Collectionwise normality and absolute retracts, Fund. Math., 1978, 98(1), 61-73; · Zbl 0391.54007
[29] Scott D., Continuous lattices, In: Toposes, Algebraic Geometry and Logic, Halifax, January 16-19, 1971, Lecture Notes in Math., 274, Springer, Berlin, 1972, 97-136 http://dx.doi.org/10.1007/BFb0073967;
[30] Stone A.H., On σ-discreteness and Borel isomorphism, Amer. J. Math., 1963, 85, 655-666 http://dx.doi.org/10.2307/2373113; · Zbl 0117.40103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.