×

On s-reflexive spaces and continuous selections. (English) Zbl 1326.54027

In this paper, the authors study s-reflexive spaces introduced by Z. Yang and D. Zhao [Fundam. Math. 192, No. 2, 111–120 (2006; Zbl 1111.47007)] in terms of continuous selections of carriers (set-valued mappings). Let \(X\) be a topological space and \(2^X\) the family of all subsets of \(X\). A family \({\mathcal A}\) of closed subsets of \(X\) is said to be reflexive if there exists a set \({\mathcal B}\) of continuous selfmaps on \(X\) such that \({\mathcal A}=\{A \in 2^X : A \text{ is closed and } f(A) \subset A \text{ for every } f \in {\mathcal B}\} \). Note that if \({\mathcal A}\) is a reflexive closed family, then \({\mathcal A}\) satisfies (a) \(X, \emptyset \in {\mathcal A}\), (b) \({\mathcal B} \subset {\mathcal A}\) implies \(\bigcap {\mathcal B} \in {\mathcal A}\), and (c) \({\mathcal B} \subset {\mathcal A}\) implies \(\overline{\bigcup {\mathcal B}} \in {\mathcal A}\). A topological space \(X\) is said to be s-reflexive if every closed family with the conditions (a)–(c) above is reflexive.
In Section 2, a characterization of s-reflexive spaces in terms of continuous selections of l.s.c.carriers is given. Applying the characterization, the authors prove, for example, that every s-reflexive Hausdorff space is zero-dimensional. In Section 3, the notion of self-selective space is introduced and discussed. A space \(X\) is said to be self-selective if every lower semicontinuous carrier \(\Phi : X \to 2^Y\) with nonempty closed values has a continuous selection. It is proved that every self-selective \(T_1\)-space is s-reflexive and countably paracompact. In Section 4, it is proved that every strongly zero-dimensional completely metrizable space is s-reflexive by applying E. Michael’s zero-dimensional selection theorem [Ann.Math.(2) 64, 562–580 (1956; Zbl 0073.17702)]. In Section 5, the s-reflexivity of several examples is discussed.

MSC:

54C60 Set-valued maps in general topology
54C65 Selections in general topology
54E35 Metric spaces, metrizability
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balogh, Z. and Rudin, M. E., Monotone normality, Topology Appl., 47, 1992, 115-127. · Zbl 0769.54022 · doi:10.1016/0166-8641(92)90066-9
[2] Boone, J. R., On irreducible spaces, Bull. Austral. Math. Soc., 12, 1975, 143-148. · Zbl 0285.54013 · doi:10.1017/S0004972700023728
[3] van Douwen, E. K., Simultaneous Extension of Continuous Functions, Doctoral Thesis, Free University, Amsterdam, 1975. · Zbl 0309.54013
[4] Engelking, R., Cartesian products and dyadic spaces, Fund. Math., 57, 1965, 287-304. · Zbl 0173.50603
[5] Engelking, R., General Topology, Polish Scientific Publishers, Warszawa, 1977. · Zbl 0373.54002
[6] Gutev, V., Private communication, Sept. 2013. · JFM 62.0228.03
[7] Gutev, V., Nedev, S., Pelant, J. and Valov, V., Cantor set selectors, Topology Appl., 44, 1992, 163-166. · Zbl 0769.54020 · doi:10.1016/0166-8641(92)90089-I
[8] Halmos, P. R., Reflexive lattices of subspaces, J. London Math. Soc., 4, 1971, 257-263. · Zbl 0231.47003 · doi:10.1112/jlms/s2-4.2.257
[9] Hausdorff, F., Summen von N1 Mengen, Fund. Math., 26, 1936, 241-255 (English translation in: J. M. Plotkin (ed.), Hausdorff on Ordered Sets, History of Math., 25, Amer. Math. Soc. and London Math. Soc., 2005). · JFM 62.0228.03
[10] Heath, R. W., Lutzer, D. J. and Zenor, P. L., Monotonically normal spaces, Trans. Amer. Math. Soc., 178, 1973, 481-493. · Zbl 0269.54009 · doi:10.1090/S0002-9947-1973-0372826-2
[11] Hurewicz, W., Relativ perfecte Teile von Punktmengen und Mengen (A), Fund. Math., 12, 1928, 78-109. · JFM 54.0097.06
[12] Jech, T., Set Theory, Academic Press, New York, 1978.
[13] Kanovei, V. G., Problem of the existence of non-Borel AF‖-sets, Math. Notes, 37(1-2), 1985, 156-161. · Zbl 0586.03039 · doi:10.1007/BF01156763
[14] Michael, E., Continuous selections I, Ann. of Math., 63, 1956, 361-382. · Zbl 0071.15902 · doi:10.2307/1969615
[15] Michael, E., Continuous selections II, Ann. of Math., 64, 1956, 562-580. · Zbl 0073.17702 · doi:10.2307/1969603
[16] Michael, E., Continuous selections and countable sets, Fund. Math., 111, 1981, 1-10. · Zbl 0455.54012
[17] Repovš, D. and Semenov, P. V., Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Netherlands, 1998. · Zbl 0915.54001 · doi:10.1007/978-94-017-1162-3
[18] Sierpiński, W., General Topology, University of Toronto Press, Toronto, 1956 (Reprinted by Dover Publications, New York, 2000). · JFM 60.0502.01
[19] Yang, Z. Q. and Zhao, D. S., Reflexive families of closed sets, Fund. Math., 192, 2006, 111-120. · Zbl 1111.47007 · doi:10.4064/fm192-2-2
[20] Yang, Z. Q. and Zhao, D. S., On reflexive closed set lattices, Comment. Math. Univ. Carolinae, 51, 2010, 143-154. · Zbl 1224.54030
[21] Zhao, D. S., On reflexive subobject lattices and reflexive endomorphism algebras, Comment. Math. Univ. Carolinae, 44, 2003, 23-32. · Zbl 1101.18303
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.