A complement theorem in the universal Menger compactum

Author:
R. B. Sher

Journal:
Proc. Amer. Math. Soc. **121** (1994), 611-618

MSC:
Primary 54C50; Secondary 55P55, 57N25

DOI:
https://doi.org/10.1090/S0002-9939-1994-1243173-0

MathSciNet review:
1243173

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Abstract | References | Similar Articles | Additional Information

Abstract: A. Chigogidze has shown that two *Z*-sets in the universal Menger compactum of dimension have the same *k*-shape if and only if their complements are homeomorphic. We show that this result holds for weak *Z*-sets. The class of weak *Z*-sets, defined herein and analogous to the weak *Z*-sets in *Q*, contains but is larger than the class of *Z*-sets. We give some examples of weak *Z*-sets in the universal Menger compactum and in *Q* that are not *Z*-sets.

**[1]**M. Bestvina,*Characterizing k-dimensional universal Menger compacta*, Mem. Amer. Math. Soc., No. 380, Amer. Math. Soc., Providence, RI, 1988. MR**920964 (89g:54083)****[2]**A. Chigogidze,*Compacta lying in the n-dimensional universal Menger compactum and having homeomorphic complements in it*, Mat. USSR-Sb.**61**(1988), 471-484. MR**911804 (89c:54071)****[3]**-, -*equivalence and n-equivalence*, Topology Appl.**45**(1992), 283-291. MR**1180815 (94e:54019)****[4]**R. J. Daverman,*Embedding phenomena based upon decomposition theory: wild Cantor sets satisfying strong homogeneity properties*, Proc. Amer. Math. Soc.**75**(1979), 177-182. MR**529237 (80k:57031)****[5]**R. J. Daverman and J. J. Walsh,*Čech homology characterizations of infinite dimensional manifolds*, Amer. J. Math.**103**(1981), 411-435. MR**618319 (83k:57008)****[6]**A. Dranishnikov,*On resolutions of*-*compacta*, Geometric Topology and Shape Theory (S. Mardešić and J. Segal, eds.), Lecture Notes in Math., vol. 1283, Springer-Verlag, New York, 1987, pp. 48-59. MR**922271 (89g:54084)****[7]**J. F. P. Hudson,*Piecewise linear topology*, W. A. Benjamin, New York, 1969. MR**0248844 (40:2094)****[8]**R. B. Sher,*Complement theorems in shape theory*, Shape Theory and Geometric Topology (S. Mardešić and J. Segal, eds.), Lecture Notes in Math., vol. 870, Springer-Verlag, New York, 1981, pp. 150-168. MR**643529 (83a:57018)****[9]**R. B. Sher,*Complement theorems in shape theory*, II, Geometric Topology and Shape Theory (S. Mardešić and J. Segal, eds.), Lecture Notes in Math., vol. 1283, Springer-Verlag, New York, 1987, pp. 212-220. MR**922283 (89d:57023)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1243173-0

Keywords:
Universal Menger compactum,
complement theorem,
*Z*-set,
weak *Z*-set,
shape theory,
*n*-shape theory

Article copyright:
© Copyright 1994
American Mathematical Society