Maximal pseudocompact spaces and the Preiss-Simon property

Ofelia Alas; Vladimir Tkachuk; Richard Wilson

Open Mathematics (2014)

  • Volume: 12, Issue: 3, page 500-509
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Ofelia Alas, Vladimir Tkachuk, and Richard Wilson. "Maximal pseudocompact spaces and the Preiss-Simon property." Open Mathematics 12.3 (2014): 500-509. <http://eudml.org/doc/268953>.

@article{OfeliaAlas2014,
abstract = {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.},
author = {Ofelia Alas, Vladimir Tkachuk, Richard Wilson},
journal = {Open Mathematics},
keywords = {Maximal pseudocompact space; Countably compact space; MP-space; Compact space; Pseudocompact space; Preiss-Simon property; maximal pseudocompact space; countably compact space; compact space; pseudocompact space},
language = {eng},
number = {3},
pages = {500-509},
title = {Maximal pseudocompact spaces and the Preiss-Simon property},
url = {http://eudml.org/doc/268953},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Ofelia Alas
AU - Vladimir Tkachuk
AU - Richard Wilson
TI - Maximal pseudocompact spaces and the Preiss-Simon property
JO - Open Mathematics
PY - 2014
VL - 12
IS - 3
SP - 500
EP - 509
AB - 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.
LA - eng
KW - Maximal pseudocompact space; Countably compact space; MP-space; Compact space; Pseudocompact space; Preiss-Simon property; maximal pseudocompact space; countably compact space; compact space; pseudocompact space
UR - http://eudml.org/doc/268953
ER -

References

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