Maximal pseudocompact spaces and the Preiss-Simon property

• Autorzy: Ofelia Alas , Vladimir Tkachuk , Richard Wilson
• Języki publikacji: 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.

Słowa kluczowe

EN

Maximal pseudocompact space; Countably compact space; MP-space; Compact space; Pseudocompact space; Preiss-Simon property

Kategorie tematyczne

• 54D99: None of the above, but in this section
• 54B10: Product spaces

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 3
• Strony: 500-509

Daty

wydano

2014-03-01

online

2013-12-21

Twórcy

autor

Ofelia Alas

• Universidade de São Paulo

autor

Vladimir Tkachuk

• Universidad Autónoma Metropolitana

autor

Richard Wilson

• Universidad Autónoma Metropolitana

Bibliografia

• [1] Arhangel’skii A.V., Structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys, 1978, 33(6), 33–96 http://dx.doi.org/10.1070/RM1978v033n06ABEH003884
• [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
• [3] Alas O.T., Sanchis M., Wilson R.G., Maximal pseudocompact and maximal R-closed spaces, Houston J. Math., 2012, 38(4), 1355–1367
• [4] Cameron D.E., A class of maximal topologies, Pacific J. Math., 1977, 70(1), 101–104 http://dx.doi.org/10.2140/pjm.1977.70.101
• [5] Efimov B.A., Dyadic bicompacta, Trudy Moskov. Mat. Obshch., 1965, 14, 211–247 (in Russian)
• [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

Identyfikatory

DOI

10.2478/s11533-013-0359-9