Definable continuous selections of set-valued maps in o-minimal expansions of the real field

• Autorzy: Sokantika S. , Thamrongthanyalak A.

Identyfikatory

DOI

10.4064/ba8130-10-2017

• Języki publikacji: EN

Abstrakty

EN

Let T be a set-valued map from a subset of Rⁿ to R^m. Suppose (R;+,⋅,T) is o-minimal. We prove that (1) if for every x ∈ Rⁿ, each connected component of T(x) is convex, then T has a continuous selection if and only if T has a continuous selection definable in (R;+,⋅,T); (2) if n = 1 or m = 1, then T has a continuous selection if and only if T has a continuous selection definable in (R;+,⋅,T).

Słowa kluczowe

EN

selections; o-minimal structures; set-valued maps

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2017
• Tom: Vol. 65, no. 2
• Strony: 97--105
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Sokantika S.

• Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

autor

Thamrongthanyalak A.

• Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Bibliografia

• [1] M. Aschenbrenner and A. Thamrongthanyalak, Whitney's extension problem in o-minimal structures, preprint, www.math.ucla.edu/~matthias/pdf/Whitney.pdf.
• [2] M. Aschenbrenner and A. Thamrongthanyalak, Michael's selection theorem in a semi-linear context, Adv. Geom. 15 (2015), 293-313.
• [3] M. Czapla and W. Pawłucki, Michael's selection theorem for a mapping definable in an o-minimal structure defined on a set of dimension 1, Topol. Methods Nonlinear Anal. 49 (2017), 377-380.
• [4] A. Fornasiero, Definably connected nonconnected sets, Math. Logic Quart. 58 (2012), 125-126.
• [5] Z. Han, X. Cai, and J. Huang, Theory of Control Systems Described by Differential Inclusions, Springer Tracts in Mechanical Engineering, Shanghai Jiaotong Univ. Press, Shanghai, and Springer, Berlin, 2016.
• [6] A. Kechris, Classical Descriptive Set Theory, Grad. Texts Math. 156, Springer, New York, 1995.
• [7] E. Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361-382.
• [8] E. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-238.
• [9] S. Park, Applications of Michael's selection theorems to fixed point theory, Topology Appl. 155 (2008), 861-870.
• [10] D. Samet, Continuous selections for vector measures, Math. Oper. Res. 12 (1987), 536-543.
• [11] A. Thamrongthanyalak, Michael's selection theorem in d-minimal expansions of the real field, preprint, pioneer.netserv.chula.ac.th/~tathipa1/dminselection.pdf.
• [12] L. van den Dries, Tame Topology and o-Minimal Structures, London Math. Soc. Lecture Note Ser. 248, Cambridge Univ. Press, Cambridge, 1998.
• [13] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).