An infinite-dimensional pre-Hilbert space not homeomorphic to its own square
Author:
Roman Pol
Journal:
Proc. Amer. Math. Soc. 90 (1984), 450-454
MSC:
Primary 57N20; Secondary 46C99, 54F45
DOI:
https://doi.org/10.1090/S0002-9939-1984-0728367-6
MathSciNet review:
728367
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Given an arbitrary infinite-dimensional separable complete linear metric space , there exists a direct sum decomposition
such that each summand
intersects every linearly independent Cantor set in
(this decomposition can be considered as a linear analogue to the classical Bernstein's decomposition into totally imperfect sets).
Theorem. Each summand of such a decomposition is not homeomorphic to its own square, and if
is a linear bounded operator, then either the kernel or the range of
is finite-dimensional.
In the case of this provides an example of a space
with the properties stated in the title, which answers a well-known question, cf. Arhangelskiĭ [A, Problem 21] and Geoghegan [G, Problem (LS 12)].
- [A] A. V. Arhangelskiĭ, Structure and classification of topological spaces, Uspehi Mat. Nauk 33 (1978), 29-84. MR 526012 (80i:54005)
- [B-P] C. Bessaga and A. Pełczyński, Selected topics in infinite-dimensional topology, PWN, Warsaw, 1975.
- [Ch] J. P. R. Christensen, Topology and Borel structure, North-Holland, Amsterdam, 1974. MR 0348724 (50:1221)
- [D] C. Dellacherie, Un cours sur les ensembles analytiques, Analytic Sets, Academic Press, London, 1980.
- [G] R. Geoghegan, Open problems in infinite-dimensional topology, Topology Proc. 4 (1979). MR 583711 (82a:57015)
- [K
] K. Kuratowski, Topology, Vols. I and II, PWN, Warsaw, 1966 and 1968. MR 0217751 (36:840)
- [K
] -, Applications of the Baire category method to the problem of independent sets, Fund. Math. 81 (1973), 65-72. MR 0339092 (49:3855)
- [M
] J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139-147. MR 0173645 (30:3855)
- [M
] -, Almost every function is independent, Fund. Math. 81 (1973), 43-48. MR 0339091 (49:3854)
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N20, 46C99, 54F45
Retrieve articles in all journals with MSC: 57N20, 46C99, 54F45
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1984-0728367-6
Keywords:
Linear metric spaces,
homeomorphisms,
Bernstein's sets
Article copyright:
© Copyright 1984
American Mathematical Society