Functor of extension in Hilbert cube and Hilbert space

Piotr Niemiec

Open Mathematics (2014)

  • Volume: 12, Issue: 6, page 887-895
  • ISSN: 2391-5455

Abstract

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It is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics.

How to cite

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Piotr Niemiec. "Functor of extension in Hilbert cube and Hilbert space." Open Mathematics 12.6 (2014): 887-895. <http://eudml.org/doc/269407>.

@article{PiotrNiemiec2014,
abstract = {It is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics.},
author = {Piotr Niemiec},
journal = {Open Mathematics},
keywords = {Z-set; Functor of extension; Hilbert cube; Fréchet space; -set; Functor},
language = {eng},
number = {6},
pages = {887-895},
title = {Functor of extension in Hilbert cube and Hilbert space},
url = {http://eudml.org/doc/269407},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Piotr Niemiec
TI - Functor of extension in Hilbert cube and Hilbert space
JO - Open Mathematics
PY - 2014
VL - 12
IS - 6
SP - 887
EP - 895
AB - It is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics.
LA - eng
KW - Z-set; Functor of extension; Hilbert cube; Fréchet space; -set; Functor
UR - http://eudml.org/doc/269407
ER -

References

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  1. [1] Anderson R.D., Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., 1966, 72(3), 515–519 http://dx.doi.org/10.1090/S0002-9904-1966-11524-0 Zbl0137.09703
  2. [2] Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383 http://dx.doi.org/10.1307/mmj/1028999787 Zbl0148.37202
  3. [3] Anderson R.D., McCharen J.D., On extending homeomorphisms to Fréchet manifolds, Proc. Amer. Math. Soc., 1970, 25(2), 283–289 Zbl0203.25805
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  5. [5] Banakh T.O., Topology of spaces of probability measures I. The functors τ τ and P ^ , Mat. Stud., 1995, 5, 65–87 (in Russian) 
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  9. [9] Chapman T.A., Deficiency in infinite-dimensional manifolds, General Topology Appl., 1971, 1, 263–272 http://dx.doi.org/10.1016/0016-660X(71)90097-3 
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  12. [12] Klee V.L. Jr., Invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc., 1952, 3(3), 484–487 http://dx.doi.org/10.1090/S0002-9939-1952-0047250-4 Zbl0047.02902
  13. [13] Kuratowski K., Mostowski A., Set Theory, 2nd ed., PWN, Warsaw, 1976 
  14. [14] Niemiec P., Spaces of measurable functions, Cent. Eur. J. Math., 2013, 11(7), 1304–1316 http://dx.doi.org/10.2478/s11533-013-0236-6 Zbl1277.54021
  15. [15] Takesaki M., Theory of Operator Algebras. I, Encyclopaedia Math. Sci., 124, Springer, Berlin, 2002 
  16. [16] Toruńczyk H., Characterizing Hilbert space topology, Fund. Math., 1981, 111(3), 247–262 Zbl0468.57015
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