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2014 | 12 | 3 | 500-509
Tytuł artykułu

Maximal pseudocompact spaces and the Preiss-Simon property

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EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
12
Numer
3
Strony
500-509
Opis fizyczny
Daty
wydano
2014-03-01
online
2013-12-21
Twórcy
autor
Bibliografia
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  • [2] Arhangel’skii A.V., Relations among the invariants of topological groups and their subspaces, Russian Math. Surveys, 1980, 35(3), 1–23 http://dx.doi.org/10.1070/RM1980v035n03ABEH001674
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  • [6] Engelking R., General Topology, IMPAN Monogr. Mat., 60, PWN, Warsaw, 1977
  • [7] Fabian M., Gâteaux Differentiability of Convex Functions and Topology, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1997
  • [8] Gillman L., Jerison M., Rings of Continuous Functions, The University Series in Higher Mathematics, Van Nostrand, Princeton, 1960 http://dx.doi.org/10.1007/978-1-4615-7819-2
  • [9] Gruenhage G., Generalized metric spaces, In: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 423–501
  • [10] Knaster B., Urbanik K., Sur les espaces complets séparables de dimension 0, Fund. Math., 1953, 40, 194–202
  • [11] Preiss D., Simon P., A weakly pseudocompact subspace of Banach space is weakly compact, Comment. Math. Univ. Carolinae, 1974, 15(4), 603–609
  • [12] Porter J.R., Stephenson R.M. Jr., Woods R.G., Maximal feebly compact spaces, Topology Appl., 1993, 52(3), 203–219 http://dx.doi.org/10.1016/0166-8641(93)90103-K
  • [13] Porter J.R., Stephenson R.M. Jr., Woods R.G., Maximal pseudocompact spaces, Comment. Math. Univ. Carolinae, 1994, 35(1), 127–145
  • [14] Reznichenko E.A., Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology Appl., 1994, 59(3), 233–244 http://dx.doi.org/10.1016/0166-8641(94)90021-3
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-013-0359-9
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