Maximal pseudocompact spaces and the Preiss-Simon property

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 com...

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Bibliographic Details
Published in:Open mathematics (Warsaw, Poland) Poland), 2014, Vol.12 (3), p.500-509
Main Authors: Alas, Ofelia, Tkachuk, Vladimir, Wilson, Richard
Format: Article
Language:English
Subjects:
54B10
54D99
Compact space
Countably compact space
Maximal pseudocompact space
MP-space
Preiss-Simon property
Pseudocompact space
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Summary: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.
ISSN:2391-5455
2391-5455
DOI:10.2478/s11533-013-0359-9