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Hyperspaces of infinite-dimensional compacta. (English) Zbl 0830.57013
The authors’ abstract: “If \(X\) is an infinite product of non-degenerate Peano continua then the set \(\dim_\infty (X)= \{A\in 2^X\): \(\dim A= \infty\}\) is an \(F_{\sigma \delta}\)-absorber in \(2^X\). As a consequence, there is a homeomorphism \(f: 2^X\to Q^\infty\) such that \(f[\dim_\infty (X) ]= B^\infty\), where \(B\) denotes the pseudo- boundary of the Hilbert cube \(Q\). There is a locally infinite-dimensional Peano continuum \(X\) such that for every \(n\), \(\dim_\infty (X^n)\) is not homeomorphic to \(B^\infty\;\)”.
Reviewer: W.Lewis (Lubbock)

MSC:
57N20 Topology of infinite-dimensional manifolds
54B20 Hyperspaces in general topology
55M10 Dimension theory in algebraic topology
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References:
[1] J. Baars , H. Gladdines , and J. Van Mill. Absorbing systems in infinite-dimensional manifolds . To appear in Top. Appl. · Zbl 0794.57005 · doi:10.1016/0166-8641(93)90019-A
[2] C. Bessaga and A. Pelczyński. Selected topics in infinite-dimensional topology . PWN, Warszawa, 1975. · Zbl 0304.57001
[3] M. Bestvina and J. Mogilski. Characterizing certain incomplete infinite-dimensional absolute retracts . Michigan Math. J., 33, 291-313, 1986. · Zbl 0629.54011 · doi:10.1307/mmj/1029003410
[4] R. Cauty. L’espace des functions continues d’un espace métrique dénombrable . Proc. Am. Math. Soc., 113, 493-501, 1991. · Zbl 0735.54008 · doi:10.2307/2048535
[5] D.W. Curtis. Hyperspaces of finite subsets as boundary sets . Top. Appl., 22, 97-107, 1986. · Zbl 0575.54009 · doi:10.1016/0166-8641(86)90081-7
[6] D.W. Curtis and M. Michael. Boundary sets for growth hyperspaces . Top. Appl., 25, 269-283, 1987. · Zbl 0627.54004 · doi:10.1016/0166-8641(87)90083-6
[7] D.W. Curtis and R.M. Schori. Hyperspaces of Peano continua are Hilbert cubes . Fund. Math., 101, 19-38, 1978. · Zbl 0409.54044 · eudml:210992
[8] J.J. Dijkstra and J. Mogilski. The topological product structure of systems of Lebesgue spaces . Math. Annalen, 290, 527-543, 1991. · Zbl 0734.46013 · doi:10.1007/BF01459258 · eudml:164831
[9] J.J. Dijkstra , J. Van Mill , and J. Mogilski. The space of infinite-dimensional compact spaces and other topological copies of (l2f)\omega . Pac. J. Math., 152, 255-273, 1992. · Zbl 0786.54012 · doi:10.2140/pjm.1992.152.255
[10] T. Dobrowolski , W. Marciszewski , and J. Mogilski. On topological classification of function spaces Cp(X) of low borel complexity . Trans. Amer. Math. Soc., 678, 307-324, 1991. · Zbl 0768.54016 · doi:10.2307/2001884
[11] A.N. Dranišnikov. On a problem of P. S. Alexandrov . Matem. Sbornik, 135, 551-557, 1988. · Zbl 0643.55001 · eudml:71964
[12] J. Dydak , J.J. Walsh. Dimension, cohomological dimension, and Sullivan’s conjecture . Preprint. · Zbl 0822.55001 · doi:10.1016/0040-9383(93)90040-3
[13] J. Van Mill. Infinite-Dimensional Topology: prerequisites and introduction . North-Holland Publishing Company, Amsterdam, 1989. · Zbl 0663.57001
[14] H. Toruńczyk. Concerning locally homotopy negligible sets and characterizations of l2manifolds . Fund. Math., 101, 93-110, 1978. · Zbl 0406.55003 · eudml:210996
[15] J.J. Walsh. Dimension, cohomological dimension and cell-like mappings . In S. Mardešić and J. Segal, editors, Shape Theory and Geometric Topology Conference, Dubrovnik , Lecture Notes in Mathematics 870, pages 105-118. Springer, Berlin, 1981. · Zbl 0474.55002
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