We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.
@article{bwmeta1.element.doi-10_2478_s11533-013-0359-9,
author = {Ofelia Alas and Vladimir Tkachuk and Richard Wilson},
title = {Maximal pseudocompact spaces and the Preiss-Simon property},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {500-509},
zbl = {1290.54013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0359-9}
}
Ofelia Alas; Vladimir Tkachuk; Richard Wilson. Maximal pseudocompact spaces and the Preiss-Simon property. Open Mathematics, Tome 12 (2014) pp. 500-509. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0359-9/
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