Michael’s Selection Theorem in d-minimal expansions of the real field
HTML articles powered by AMS MathViewer
- by Athipat Thamrongthanyalak PDF
- Proc. Amer. Math. Soc. 147 (2019), 1059-1071 Request permission
Abstract:
Let $E \subseteq \mathbb {R}^n$. If $T$ is a lower semi-continuous set-valued map from $E$ to $\mathbb {R}^m$ and $(\mathbb {R},+,\cdot ,T)$ is d-minimal, then there is a continuous function $f \colon E \to \mathbb {R}^m$ definable in $(\mathbb {R},+,\cdot ,T)$ such that $f(x) \in T(x)$ for every $x \in E$. To prove this result, we establish a cell decomposition theorem for d-minimal expansions of the real field.References
- Matthias Aschenbrenner and Andreas Fischer, Definable versions of theorems by Kirszbraun and Helly, Proc. Lond. Math. Soc. (3) 102 (2011), no. 3, 468–502. MR 2783134, DOI 10.1112/plms/pdq029
- M. Aschenbrenner and A. Thamrongthanyalak, Whitney’s Extension Problem in o-minimal structures (2013), To appear in Revista Matemática Iberoamericana, available at www.math.ucla.edu/~matthias/pdf/Whitney.pdf.
- Matthias Aschenbrenner and Athipat Thamrongthanyalak, Michael’s selection theorem in a semilinear context, Adv. Geom. 15 (2015), no. 3, 293–313. MR 3365747, DOI 10.1515/advgeom-2015-0018
- Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
- Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337, DOI 10.1215/S0012-7094-96-08416-1
- Harvey Friedman and Chris Miller, Expansions of o-minimal structures by sparse sets, Fund. Math. 167 (2001), no. 1, 55–64. MR 1816817, DOI 10.4064/fm167-1-4
- Harvey Friedman and Chris Miller, Expansions of o-minimal structures by fast sequences, J. Symbolic Logic 70 (2005), no. 2, 410–418. MR 2140038, DOI 10.2178/jsl/1120224720
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
- Ernest Michael, Continuous selections. II, Ann. of Math. (2) 64 (1956), 562–580. MR 80909, DOI 10.2307/1969603
- Chris Miller, Tameness in expansions of the real field, Logic Colloquium ’01, Lect. Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 281–316. MR 2143901
- C. Miller, Definable choice in d-minimal expansions of ordered groups (2006), unpublished note, available at https:// people.math.osu.edu/miller.1987/eidmin.pdf.
- Chris Miller, Expansions of o-minimal structures on the real field by trajectories of linear vector fields, Proc. Amer. Math. Soc. 139 (2011), no. 1, 319–330. MR 2729094, DOI 10.1090/S0002-9939-2010-10506-3
- C. Miller and A. Thamrongthanyalak, D-minimal expansions of the real field have the $C^p$ zero set property, Proc. Amer. Soc. 146 (2018), no. 12, 5169-5179.
- Chris Miller and James Tyne, Expansions of o-minimal structures by iteration sequences, Notre Dame J. Formal Logic 47 (2006), no. 1, 93–99. MR 2211185, DOI 10.1305/ndjfl/1143468314
- James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
- Sehie Park, Applications of Michael’s selection theorems to fixed point theory, Topology Appl. 155 (2008), no. 8, 861–870. MR 2406393, DOI 10.1016/j.topol.2007.02.017
- M. Zippin, Applications of Michael’s continuous selection theorem to operator extension problems, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1371–1378. MR 1487350, DOI 10.1090/S0002-9939-99-04777-2
Additional Information
- Athipat Thamrongthanyalak
- Affiliation: Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10400, Thailand
- Email: athipat.th@chula.ac.th
- Received by editor(s): March 8, 2017
- Received by editor(s) in revised form: March 5, 2018
- Published electronically: December 3, 2018
- Communicated by: Heike Mildenberger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1059-1071
- MSC (2010): Primary 26B05; Secondary 03C64
- DOI: https://doi.org/10.1090/proc/14283
- MathSciNet review: 3896056