A+ CATEGORY SCIENTIFIC UNIT

Regular spaces of small extent are $\omega $-resolvable

Volume 228 / 2015

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy Fundamenta Mathematicae 228 (2015), 27-46 MSC: 54A35, 03E35, 54A25. DOI: 10.4064/fm228-1-3

Abstract

We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space $X$ that satisfies $\varDelta (X)>\operatorname {\rm e}(X)$ is ${\omega }$-resolvable. Here $\varDelta (X)$, the dispersion character of $X$, is the smallest size of a non-empty open set in $X$, and $\operatorname {\rm e}(X)$, the extent of $X$, is the supremum of the sizes of all closed-and-discrete subsets of $X$. In particular, regular Lindelöf spaces of uncountable dispersion character are ${\omega }$-resolvable.

We also prove that any regular Lindelöf space $X$ with $|X|=\varDelta (X)=\omega _1$ is even ${\omega _1}$-resolvable. The question whether regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.

Authors

  • István JuhászAlfréd Rényi Institute of Mathematics
    Hungarian Academy of Sciences
    13–15 Reáltanoda u.
    1053 Budapest, Hungary
    e-mail
  • Lajos SoukupAlfréd Rényi Institute of Mathematics
    Hungarian Academy of Sciences
    13–15 Reáltanoda u.
    1053 Budapest, Hungary
    e-mail
  • Zoltán SzentmiklóssyInstitute of Mathematics
    Faculty of Science
    Eötvös Loránd University
    Pázmány Péter sétány 1/C
    1117 Budapest, Hungary
    e-mail

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