Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Stability and dimension—a counterexample to a conjecture of Chogoshvili
HTML articles powered by AMS MathViewer

by Yaki Sternfeld PDF
Trans. Amer. Math. Soc. 340 (1993), 243-251 Request permission

Abstract:

For every $n \geq 2$ we construct an $n$-dimensional compact subset $X$ of some Euclidean space $E$ so that none of the canonical projections of $E$ on its two-dimensional coordinate subspaces has a stable value when restricted to $X$. This refutes a longstanding claim due to Chogoshvili. To obtain this we study the lattice of upper semicontinuous decompositions of $X$ and in particular its sublattice that consists of monotone decompositions when $X$ is hereditarily indecomposable.
References
  • R. H. Bing, Higher-dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 267–273. MR 43452, DOI 10.1090/S0002-9947-1951-0043452-5
  • C. Bessaga and A. Pełczyński, Selected topics in infinite dimensional topology, PWN, Warsaw, 1975.
  • George Chogoshvili, On a theorem in the theory of dimensionality, Compositio Math. 5 (1938), 292–298. MR 1556998
  • R.. Engelking, Math. Rev. 90:k 54047. —, Dimension theory, North-Holland, 1978.
  • Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
  • K. Kuratowski, Topology. II, Academic Press and PWN, 1968. R. Pol, A $2$-dimensional compactum in the product of two $1$-dimensional compacta which does not contain any rectangle, Ulam Quarterly (to appear).
  • K. Sitnikov, Example of a two-dimensional set in three-dimensional Euclidean space allowing arbitrarily small deformations into a one-dimensional polyhedron and a certain new characteristic of the dimension of sets in Euclidean spaces, Doklady Akad. Nauk SSSR (N.S.) 88 (1953), 21–24 (Russian). MR 0054245
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 54F45
  • Retrieve articles in all journals with MSC: 54F45
Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 340 (1993), 243-251
  • MSC: Primary 54F45
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1145964-7
  • MathSciNet review: 1145964