On infinite-dimensional manifold triples
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- by Katsuro Sakai and Raymond Y. Wong PDF
- Trans. Amer. Math. Soc. 318 (1990), 545-555 Request permission
Abstract:
Let $Q$ denote the Hilbert cube ${[ - 1,1]^\omega },\;s = {( - 1,1)^\omega }$ the pseudo-interior of $Q,\;\Sigma = \{ ({x_i}) \in s|\sup |{x_i}| < 1\}$ and $\sigma = \{ ({x_i}) \in s|{x_i} = 0\;{\text {except for finitely many}}\;i\}$. A triple $(X,M,N)$ of separable metrizable spaces is called a $(Q,\Sigma ,\sigma )$- (or $(s,\Sigma ,\sigma )$-)manifold triple if it is locally homeomorphic to $(Q,\Sigma ,\sigma )$ (or $(s,\Sigma ,\sigma )$). In this paper, we study such manifold triples and give some characterizations.References
- R. D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515–519. MR 190888, DOI 10.1090/S0002-9904-1966-11524-0
- R. D. Anderson, Strongly negligible sets in Fréchet manifolds, Bull. Amer. Math. Soc. 75 (1969), 64–67. MR 238358, DOI 10.1090/S0002-9904-1969-12146-4 —, On sigma-compact subsets of infinite-dimensional spaces, unpublished manuscript.
- R. D. Anderson and T. A. Chapman, Extending homeomorphisms to Hilbert cube manifolds, Pacific J. Math. 38 (1971), 281–293. MR 319204
- R. D. Anderson, David W. Henderson, and James E. West, Negligible subsets of infinite-dimensional manifolds, Compositio Math. 21 (1969), 143–150. MR 246326
- R. D. Anderson and John D. McCharen, On extending homeomorphisms to Fréchet manifolds, Proc. Amer. Math. Soc. 25 (1970), 283–289. MR 258064, DOI 10.1090/S0002-9939-1970-0258064-5
- R. D. Anderson and R. Schori, Factors of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 142 (1969), 315–330. MR 246327, DOI 10.1090/S0002-9947-1969-0246327-5
- C. Bessaga and A. Pełczyński, The estimated extension theorem, homogeneous collections and skeletons, and their applications to the topological classification of linear metric spaces and convex sets, Fund. Math. 69 (1970), 153–190. MR 273347, DOI 10.4064/fm-69-2-153-190 —, Selected topics in infinite-dimensional topology, Monogr. Mat. 58, PWN, Warsaw, 1970.
- Mladen Bestvina and Jerzy Mogilski, Characterizing certain incomplete infinite-dimensional absolute retracts, Michigan Math. J. 33 (1986), no. 3, 291–313. MR 856522, DOI 10.1307/mmj/1029003410
- T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399–426. MR 283828, DOI 10.1090/S0002-9947-1971-0283828-7
- T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR 0423357
- D. W. Curtis, Boundary sets in the Hilbert cube, Topology Appl. 20 (1985), no. 3, 201–221. MR 804034, DOI 10.1016/0166-8641(85)90089-6
- Doug Curtis, Tadeusz Dobrowolski, and Jerzy Mogilski, Some applications of the topological characterizations of the sigma-compact spaces $l^{2}_{f}$ and $\Sigma$, Trans. Amer. Math. Soc. 284 (1984), no. 2, 837–846. MR 743748, DOI 10.1090/S0002-9947-1984-0743748-7
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- Steve Ferry, The homeomorphism group of a compact Hilbert cube manifold is an $\textrm {ANR}$, Ann. of Math. (2) 106 (1977), no. 1, 101–119. MR 461536, DOI 10.2307/1971161
- Ross Geoghegan, On spaces of homeomorphisms, embeddings, and functions. II. The piecewise linear case, Proc. London Math. Soc. (3) 27 (1973), 463–483. MR 328969, DOI 10.1112/plms/s3-27.3.463
- David W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25–33. MR 250342, DOI 10.1016/0040-9383(70)90046-7
- Jerzy Mogilski, Characterizing the topology of infinite-dimensional $\sigma$-compact manifolds, Proc. Amer. Math. Soc. 92 (1984), no. 1, 111–118. MR 749902, DOI 10.1090/S0002-9939-1984-0749902-8
- Katsuro Sakai, On topologies of triangulated infinite-dimensional manifolds, J. Math. Soc. Japan 39 (1987), no. 2, 287–300. MR 879930, DOI 10.2969/jmsj/03920287 —, A $Q$-manifold local-compactification of metric combinatorial $\infty$-manifold, Proc. Amer. Math. Soc. 100 (1987). —, The space of Lipschitz maps from a compactum to a polyhedron, in preparation.
- Katsuro Sakai and Raymond Y. Wong, The space of Lipschitz maps from a compactum to a locally convex set, Topology Appl. 32 (1989), no. 3, 223–235. MR 1007102, DOI 10.1016/0166-8641(89)90030-8
- James E. West, Infinite products which are Hilbert cubes, Trans. Amer. Math. Soc. 150 (1970), 1–25. MR 266147, DOI 10.1090/S0002-9947-1970-0266147-3
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 318 (1990), 545-555
- MSC: Primary 57N20; Secondary 58B99
- DOI: https://doi.org/10.1090/S0002-9947-1990-0994171-4
- MathSciNet review: 994171