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On Chogoshvili's conjecture

Author: A. N. Dranishnikov
Journal: Proc. Amer. Math. Soc. 125 (1997), 2155-2160
MSC (1991): Primary 55M10, 54F45
MathSciNet review: 1425119
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Abstract: There exists a two-dimensional compact subset of $\mathbb {R}^{4}$ having unstable intersection with every affine 2-plane.

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  • 1. P. Alexandroff, Zum allgemeinen Dimensionsproblem, Gott. Nachrichten 37 (1928).
  • 2. G. Chogoshvili, On a theorem in the theory of dimensionality, Compositio Math. 5 (1938), 292-298.
  • 3. K.A. Sitnikov, An example of a two-dimensional set in three-dimensional Euclidean space which does not separate any regions of that space (in Russian), Dokl. Akad. Nauk SSSR 94 (1954), 1007-1010.
  • 4. G. Nobeling, Die Projektioner einer kompakten m-dimensionalen Menge in $R_{k}$, Ergebnisse Math. Kolloq. 4 (1933), 24-25.
  • 5. S. Mardesic, Compact subsets of ${\mathbb R}^{n}$ and dimension of their projections, Proc. Amer. Math. Soc. 41 (1973), 631-633.
  • 6. M. Levin, A proof of Chogoshvili conjecture for some 2-dimensional compacta, Preprint (1995).
  • 7. D.O. Kiguradze, Some properties of metric dimension (in Russian), Soobsch. Akad. Nauk. Gruz SSR 132:3 (1988), 485-488.
  • 8. Y. Sternfeld, Stability and Dimension - a counterexample to a conjecture of Chogoshvili, Transactions AMS 340 (1) (1993).
  • 9. F. Ancel and T. Dobrowolski, On the Sternfel-Levin Counterexamples to a conjecture of Chogoshvili-Pontryagin, Preprint.
  • 10. A. Dranishnikov, D. Repovs and E. Schepin, On intersection of compacta in euclidean space: the metastable case, Tsukuba J. Math. 17:2 (1993), 549-564.
  • 11. E.M. Chirka, Complex Analytic Sets, Kluwer, Academic Publish, 1989.

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Additional Information

A. N. Dranishnikov
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

Received by editor(s): May 15, 1995
Additional Notes: Partially supported by NSF grant DMS-9500875.
Communicated by: James E. West
Article copyright: © Copyright 1997 American Mathematical Society

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