Characterizing the topology of infinite-dimensional 𝜎-compact manifolds

Mogilski, Jerzy

doi:10.1090/S0002-9939-1984-0749902-8

Characterizing the topology of infinite-dimensional $\sigma$-compact manifolds
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Proc. Amer. Math. Soc. 92 (1984), 111-118 Request permission

Abstract:

A metric space $(X,d)$, which is a countable union of finite-dimensional compacta, is a manifold modelled on the space $l_2^f = \{ ({x_i}) \in {l_2}$ all but finitely many ${x_i} = 0\}$ iff $X$ is an ANR and the following condition holds: given $\varepsilon > 0$, a pair of finite-dimensional compacta $(A,B)$ and a map $f:A \to X$ such that $f|B$ is an embedding, there is an embedding $g:A \to X$ such that $g\left | {B = f} \right |B$ and $d(f(x),g(x)) < \varepsilon$ for all $x \in A$. An analogous condition characterizes manifolds modelled on the space $\Sigma = \{ ({x_i}) \in {l_2}:\sum _{i = 1}^\infty {(i{x_i})^2} < \infty \}$.

References

On sigma-compact subsets of finite-dimensional spaces

R. D. Anderson, D. W. Curtis, and J. van Mill, A fake topological Hilbert space, Trans. Amer. Math. Soc. 272 (1982), no. 1, 311–321. MR 656491, DOI 10.1090/S0002-9947-1982-0656491-8
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

D. W. Curtis, Hyperspaces of finite subsets as boundary sets, Topology Appl. 22 (1986), no. 1, 97–107. MR 831185, DOI 10.1016/0166-8641(86)90081-7

Some applications of the topological characterizations of the sigma-compact spaces

T. Dobrowolski and J. Mogilski, Sigma-compact locally convex metric linear spaces universal for compacta are homeomorphic, Proc. Amer. Math. Soc. 85 (1982), no. 4, 653–658. MR 660623, DOI 10.1090/S0002-9939-1982-0660623-0
Ross Geoghegan, Manifolds of piecewise linear maps and a related normed linear space, Bull. Amer. Math. Soc. 77 (1971), 629–632. MR 277010, DOI 10.1090/S0002-9904-1971-12782-9
Open problems in infinite-dimensional topology, Topology Proc. 4 (1979), no. 1, 287–338 (1980). MR 583711
Ross Geoghegan and William E. Haver, On the space of piecewise linear homeomorphisms of a manifold, Proc. Amer. Math. Soc. 55 (1976), no. 1, 145–151. MR 402785, DOI 10.1090/S0002-9939-1976-0402785-3
James P. Henderson and John J. Walsh, Examples of cell-like decompositions of the infinite-dimensional manifolds $\sigma$ and $\Sigma$, Topology Appl. 16 (1983), no. 2, 143–154. MR 712860, DOI 10.1016/0166-8641(83)90014-7
Jerzy Mogilski, CE-decomposition of $l_{2}$-manifolds, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 3-4, 309–314 (English, with Russian summary). MR 552055

Cell-like decompositions of ANR’s

Skeletons and absorbing sets in complete meric spaces

H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of $l_{2}$-manifolds, Fund. Math. 101 (1978), no. 2, 93–110. MR 518344, DOI 10.4064/fm-101-2-93-110

An introduction to infinite-dimensional topology

H. Toruńczyk, Absolute retracts as factors of normed linear spaces, Fund. Math. 86 (1974), 53–67. MR 365471, DOI 10.4064/fm-86-1-53-67
H. Toruńczyk, On $\textrm {CE}$-images of the Hilbert cube and characterization of $Q$-manifolds, Fund. Math. 106 (1980), no. 1, 31–40. MR 585543, DOI 10.4064/fm-106-1-31-40
H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247–262. MR 611763, DOI 10.4064/fm-111-3-247-262
James E. West, The ambient homeomorphy of an incomplete subspace of infinite-dimensional Hilbert spaces, Pacific J. Math. 34 (1970), 257–267. MR 277011, DOI 10.2140/pjm.1970.34.257

Characterizing Hilbert space manifolds revisited