Reflecting topological properties in continuous images

Vladimir Tkachuk

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 456-465
  • ISSN: 2391-5455

Abstract

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Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ + for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ + for compact spaces. We present examples showing that separability, countable extent and normality do not reflect in continuous images of weight ω 1. Besides, under MA + ¬ CH, the Fréchet-Urysohn property does not reflect in continuous images of weight ω 1 even for compact spaces. An application of our techniques gives a solution of an open problem published by Ramírez-Páramo. If Jensen’s κ +-Axiom ♦ κ + holds for an infinite cardinal κ, then for an arbitrary space X with no G κ-points there exists a continuous surjective map f: X → Y such that w(Y) = κ + and Y has no G tk-points. We apply this result to solve a problem of Kalenda.

How to cite

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Vladimir Tkachuk. "Reflecting topological properties in continuous images." Open Mathematics 10.2 (2012): 456-465. <http://eudml.org/doc/269228>.

@article{VladimirTkachuk2012,
abstract = {Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ + for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ + for compact spaces. We present examples showing that separability, countable extent and normality do not reflect in continuous images of weight ω 1. Besides, under MA + ¬ CH, the Fréchet-Urysohn property does not reflect in continuous images of weight ω 1 even for compact spaces. An application of our techniques gives a solution of an open problem published by Ramírez-Páramo. If Jensen’s κ +-Axiom \[\left( \{\diamondsuit \_\{\kappa ^ + \} \} \right)\] holds for an infinite cardinal κ, then for an arbitrary space X with no G κ-points there exists a continuous surjective map f: X → Y such that w(Y) = κ + and Y has no G tk-points. We apply this result to solve a problem of Kalenda.},
author = {Vladimir Tkachuk},
journal = {Open Mathematics},
keywords = {Continuous images; Reflected properties; Small weight; Jensen’s Axiom; Diamond principle; Space with no G K-points; Spread; Extent; Souslin number; Weight; Network weight; Corson space; Character; π-character; continuous images; reflected properties; small weight; character; pseudocharacter; network weight},
language = {eng},
number = {2},
pages = {456-465},
title = {Reflecting topological properties in continuous images},
url = {http://eudml.org/doc/269228},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Vladimir Tkachuk
TI - Reflecting topological properties in continuous images
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 456
EP - 465
AB - Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ + for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ + for compact spaces. We present examples showing that separability, countable extent and normality do not reflect in continuous images of weight ω 1. Besides, under MA + ¬ CH, the Fréchet-Urysohn property does not reflect in continuous images of weight ω 1 even for compact spaces. An application of our techniques gives a solution of an open problem published by Ramírez-Páramo. If Jensen’s κ +-Axiom \[\left( {\diamondsuit _{\kappa ^ + } } \right)\] holds for an infinite cardinal κ, then for an arbitrary space X with no G κ-points there exists a continuous surjective map f: X → Y such that w(Y) = κ + and Y has no G tk-points. We apply this result to solve a problem of Kalenda.
LA - eng
KW - Continuous images; Reflected properties; Small weight; Jensen’s Axiom; Diamond principle; Space with no G K-points; Spread; Extent; Souslin number; Weight; Network weight; Corson space; Character; π-character; continuous images; reflected properties; small weight; character; pseudocharacter; network weight
UR - http://eudml.org/doc/269228
ER -

References

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  12. [12] Tkachenko M.G., Continuous mappings onto spaces of smaller weight, Moscow Univ. Math. Bull., 1980, 35(2), 41–44 Zbl0459.54003
  13. [13] Tkachuk V.V., Spaces that are projective with respect to classes of mappings, Trans. Moscow Math. Soc., 1988, 139–156 Zbl0662.54007
  14. [14] Tkachuk V.V., A short proof of a classical result of M.G. Tkachenko, Topology Proc., 2001/02, 26(2), 851–856 Zbl1083.54017
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