A complement theorem in the universal Menger compactum

Sher, R. B.

doi:10.1090/S0002-9939-1994-1243173-0

A complement theorem in the universal Menger compactum
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Proc. Amer. Math. Soc. 121 (1994), 611-618 Request permission

Abstract:

A. Chigogidze has shown that two Z-sets in the universal Menger compactum of dimension $k + 1$ 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.

References

Mladen Bestvina, Characterizing $k$-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (1988), no. 380, vi+110. MR 920964, DOI 10.1090/memo/0380
A. Ch. Chigogidze, Compact spaces lying in the $n$-dimensional universal Menger compact space and having homeomorphic complements in it, Mat. Sb. (N.S.) 133(175) (1987), no. 4, 481–496, 559 (Russian); English transl., Math. USSR-Sb. 61 (1988), no. 2, 471–484. MR 911804, DOI 10.1070/SM1988v061n02ABEH003219
A. Chigogidze, $UV^n$-equivalence and $n$-equivalence, Proceedings of the Tsukuba Topology Symposium (Tsukuba, 1990), 1992, pp. 283–291. MR 1180815, DOI 10.1016/0166-8641(92)90010-W
Robert J. Daverman, Embedding phenomena based upon decomposition theory: wild Cantor sets satisfying strong homogeneity properties, Proc. Amer. Math. Soc. 75 (1979), no. 1, 177–182. MR 529237, DOI 10.1090/S0002-9939-1979-0529237-7
Robert J. Daverman and John J. Walsh, Čech homology characterizations of infinite-dimensional manifolds, Amer. J. Math. 103 (1981), no. 3, 411–435. MR 618319, DOI 10.2307/2374099
A. N. Dranishnikov, On resolutions of $\textrm {LC}^n$-compacta, Geometric topology and shape theory (Dubrovnik, 1986) Lecture Notes in Math., vol. 1283, Springer, Berlin, 1987, pp. 48–59. MR 922271, DOI 10.1007/BFb0081418
J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR 0248844
R. B. Sher, Complement theorems in shape theory, Shape theory and geometric topology (Dubrovnik, 1981) Lecture Notes in Math., vol. 870, Springer, Berlin-New York, 1981, pp. 150–168. MR 643529
R. B. Sher, Complement theorems in shape theory. II, Geometric topology and shape theory (Dubrovnik, 1986) Lecture Notes in Math., vol. 1283, Springer, Berlin, 1987, pp. 212–220. MR 922283, DOI 10.1007/BFb0081430