Stability and dimension—a counterexample to a conjecture of Chogoshvili
Author:
Yaki Sternfeld
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
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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.
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- 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
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© Copyright 1993
American Mathematical Society