# 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

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topOfelia 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 -

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