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Keywords:
Hilbert space; Hilbert cube; $\Cal F_{\sigma\delta}$-absorber; ambient homeomorphism; function space; $p$-summable sequence
Hilbert space; Hilbert cube; $\Cal F_{\sigma\delta}$-absorber; ambient homeomorphism; function space; $p$-summable sequence
Summary:
In this paper we consider a number of sequence and function spaces that are known to be homeomorphic to the countable product of the linear space $\sigma$. The spaces we are interested in have a canonical imbedding in both a topological Hilbert space and a Hilbert cube. It turns out that when we consider these spaces as subsets of a Hilbert cube then there is only one topological type. For imbeddings in the countable product of lines there are two types depending on whether the space is contained in a $\sigma$-compactum or not.
References:
[1] Baars J., Gladdines H., van Mill J.: Absorbing systems in infinite-dimensional manifolds. Topology Appl. 50 (1993), 147-182. MR 1217483 | Zbl 0794.57005
[2] Bessaga C., Pełczyński A.: Selected Topics in Infinite-Dimensional Topology. PWN Warsaw (1975).
[3] Curtis D.W.: Boundary sets in the Hilbert cube. Topology Appl. 20 (1985), 201-221. MR 0804034 | Zbl 0575.57008
[4] Dijkstra J.J., Dobrowolski T., Marciszewski W., van Mill J., Mogilski J.: Recent classification and characterization results in geometric topology. Bull. Amer. Math. Soc. 22 (1990), 277-283. MR 1027899 | Zbl 0713.57011
[5] Dijkstra J.J., van Mill J., Mogilski J.: The space of infinite-dimensional compacta and other topological copies of $(l^2_{an f})^ømega$. Pacific. J. Math. 152 (1992), 255-273. MR 1141795
[6] Dijkstra J.J., Mogilski J.: The topological product structure of systems of Lebesgue spaces. Math. Ann. 290 (1991), 527-543. MR 1116236 | Zbl 0734.46013
[7] Dobrowolski T., Marciszewski W., Mogilski J.: On topological classification of function spaces $C_{an p}(X)$ of low Borel complexity. Trans. Amer. Math. Soc. 328 (1991), 307-324. MR 1065602
[8] Dobrowolski T., Mogilski J.: Problems on topological classification of incomplete metric spaces. 409-429 in Open Problems in Topology, J. van Mill and G.M. Reed, eds., North-Holland, Amsterdam, 1990. MR 1078661
[9] Dobrowolski T., Mogilski J.: Certain sequence and function spaces homeomorphic to the countable product of $l_f^2$. J. London Math. Soc. 45.2 (1992), 566-576. MR 1180263
[10] Lutzer D., McCoy R.: Category in function spaces I. Pacific J. Math. 90 (1980), 145-168. MR 0599327 | Zbl 0481.54017
[11] van Mill J.: Topological equivalence of certain function spaces. Compositio Math. 63 (1987), 159-188. MR 0906368 | Zbl 0634.54011
[12] van Mill J.: Infinite-Dimensional Topology, Prerequisites and Introduction. North-Holland Amsterdam (1989). MR 0977744 | Zbl 0663.57001
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