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 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.
@article{bwmeta1.element.doi-10_2478_s11533-011-0141-9,
author = {Vladimir Tkachuk},
title = {Reflecting topological properties in continuous images},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {456-465},
zbl = {1257.54022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0141-9}
}
Vladimir Tkachuk. Reflecting topological properties in continuous images. Open Mathematics, Tome 10 (2012) pp. 456-465. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0141-9/
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