The Lindelöf number of C p(X)×C p(X) for strongly zero-dimensional X

Oleg Okunev

Open Mathematics (2011)

  • Volume: 9, Issue: 5, page 978-983
  • ISSN: 2391-5455

Abstract

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We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C p(X, M) is a continuous image of a closed subspace of C p(X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindelöf number of C p(X)×C p(X) coincides with the Lindelöf number of C p(X). We also prove that l(C p(X n)κ) ≤ l(C p(X)κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.

How to cite

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Oleg Okunev. "The Lindelöf number of C p(X)×C p(X) for strongly zero-dimensional X." Open Mathematics 9.5 (2011): 978-983. <http://eudml.org/doc/269680>.

@article{OlegOkunev2011,
abstract = {We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C p(X, M) is a continuous image of a closed subspace of C p(X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindelöf number of C p(X)×C p(X) coincides with the Lindelöf number of C p(X). We also prove that l(C p(X n)κ) ≤ l(C p(X)κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.},
author = {Oleg Okunev},
journal = {Open Mathematics},
keywords = {Topology of pointwise convergence; Lindelöf number; topology of pointwise convergence; (strongly) zero-dimensional spaces},
language = {eng},
number = {5},
pages = {978-983},
title = {The Lindelöf number of C p(X)×C p(X) for strongly zero-dimensional X},
url = {http://eudml.org/doc/269680},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Oleg Okunev
TI - The Lindelöf number of C p(X)×C p(X) for strongly zero-dimensional X
JO - Open Mathematics
PY - 2011
VL - 9
IS - 5
SP - 978
EP - 983
AB - We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C p(X, M) is a continuous image of a closed subspace of C p(X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindelöf number of C p(X)×C p(X) coincides with the Lindelöf number of C p(X). We also prove that l(C p(X n)κ) ≤ l(C p(X)κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.
LA - eng
KW - Topology of pointwise convergence; Lindelöf number; topology of pointwise convergence; (strongly) zero-dimensional spaces
UR - http://eudml.org/doc/269680
ER -

References

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  1. [1] Arhangel’skiĭ A.V., Problems in C p-theory, In: Open Problems in Topology, North-Holland, Amsterdam, 1990, 601–615 
  2. [2] Arhangel’skiĭ A.V., Topological Function Spaces, Math. Appl. (Soviet Ser.), 78, Kluwer, Dordrecht, 1992 
  3. [3] Engelking R., General Topology, Sigma Ser. Pure Math., 6, Heldermann, Berlin, 1989 
  4. [4] Mardešić S., On covering dimension and inverse limits of compact spaces, Illinois J. Math, 1960, 4(2), 278–291 Zbl0094.16902
  5. [5] Okunev O., On Lindelöf Σ-spaces of continuous functions in the pointwise topology, Topology Appl., 1993, 49(2), 149–166 http://dx.doi.org/10.1016/0166-8641(93)90041-B 
  6. [6] Okunev O., Tamano K., Lindelöf powers and products of function spaces, Proc. Amer. Math. Soc., 1996, 124(9), 2905–2916 http://dx.doi.org/10.1090/S0002-9939-96-03629-5 Zbl0858.54013
  7. [7] Tkachuk V.V., Some criteria for C p(X) to be an LΣ(≤ ω)-space (in preparation) 

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