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 we construct an -dimensional compact subset of some Euclidean space so that none of the canonical projections of on its two-dimensional coordinate subspaces has a stable value when restricted to . This refutes a longstanding claim due to Chogoshvili. To obtain this we study the lattice of upper semicontinuous decompositions of and in particular its sublattice that consists of monotone decompositions when is hereditarily indecomposable.

**[B]**R. H. Bing,*Higher dimensional hereditarily indecomposable continua*, Trans. Amer. Math. Soc.**71**(1951), 267-273. MR**0043452 (13:265c)****[B.P]**C. Bessaga and A. Pełczyński,*Selected topics in infinite dimensional topology*, PWN, Warsaw, 1975.**[Ch]**G. Chogoshvili,*On a theorem in the theory of dimensionality*, Compositio Math**5**(1938), 292-298. MR**1556998****[E]**R.. Engelking, Math. Rev.**90:k**54047.**[E]**-,*Dimension theory*, North-Holland, 1978.**[H.W]**W. Hurewicz and H. Wallman,*Dimension theory*, Princeton Univ. Press, Princeton, N.J., 1941. MR**0006493 (3:312b)****[Ku]**K. Kuratowski,*Topology*. II, Academic Press and PWN, 1968.**[P]**R. Pol,*A*-*dimensional compactum in the product of two*-*dimensional compacta which does not contain any rectangle*, Ulam Quarterly (to appear).**[Si]**K. Sitnikov,*An example of two-dimensional set in three-dimensional Euclidean space allowing arbitrarily small deformations into a one- dimensional polyhedron and a certain new characterization of the dimension of sets in Euclidean spaces*, Dokl. Akad. Nauk SSSR**88**(1953), 21-24. MR**0054245 (14:894b)**

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DOI:
https://doi.org/10.1090/S0002-9947-1993-1145964-7

Article copyright:
© Copyright 1993
American Mathematical Society