On Small Subsets in Euclidean Spaces

• Autorzy: Mycielski J. , Tomkowicz G.

Identyfikatory

DOI

10.4064/ba8085-10-2016

• Języki publikacji: EN

Abstrakty

EN

We study a property of smallness of sets which is stronger than the possibility of packing the set into arbitrarily small balls (i.e., being Tarski null) but weaker than paradoxical decomposability (i.e., being a disjoint union of two sets equivalent by finite decomposition to the whole). We show, using the Axiom of Choice for uncountable families, that there are Tarski null sets which are not small sets. Using only the Principle of Dependent Choices, we show that bounded subsets of Rⁿ that are included in countable unions of proper analytic subsets of Rⁿ are small, and several related results.

Słowa kluczowe

EN

analytic set; equivalence by finite decomposition; small set; Tarski absolute measure

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2016
• Tom: Vol. 64, no. 2/3
• Strony: 109--118
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Mycielski J.

• Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, U.S.A.

autor

Tomkowicz G.

• Centrum Edukacji G2, Moniuszki 9, 41-902 Bytom, Poland

Bibliografia

• [1] S. Banach et A. Tarski, Sur la décomposition des ensembles de points en parties respectivement congruents, Fund. Math. 6 (1924), 244–277.
• [2] M. R. Burke, Paradoxical decompositions of planar sets of positive outer measure, J. Geom. 79 (2004), 56–58.
• [3] H. Hadwiger, Absolut messbare Punktmengen im euklidischen Raum, Comment. Math. Helv. 28 (1954), 119–148.
• [4] J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147.
• [5] J. Mycielski, The Banach–Tarski paradox for the hyperbolic plane, Fund. Math. 132 (1989), 143–149.
• [6] J. Mycielski and G. Tomkowicz, The Banach–Tarski paradox for the hyperbolic plane (II), Fund. Math. 222 (2013), 289–290.
• [7] J. Mycielski and G. Tomkowicz, Shadows of the Axiom Choice in L(R), in preparation.
• [8] K. J. Nowak, A theorem on generic intersections in an o-minimal structure, Fund. Math. 227 (2014), 21–25.
• [9] K. J. Nowak and G. Tomkowicz, Intersection of generic rotations in some classical spaces, Bull. Polish Acad. Sci. Math. 64 (2016), 105–107.
• [10] W. Sierpiński, On the Congruence of Sets and Their Equivalence by Finite Decomposition, Lucknow Univ., 1954; reprinted by Chelsea, 1967.
• [11] A. Tarski, Über das absolute Mass linearer Punktmengen, Fund. Math. 30 (1938), 218–234.
• [12] A. Tarski, Algebraische Fassung des Massproblems, Fund. Math. 31 (1938), 47–66.
• [13] G. Tomkowicz and S. Wagon, The Banach–Tarski Paradox, 2nd ed., Cambridge Univ. Press, 2016.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.