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

István Juhász¹ ; Lajos Soukup¹ ; Zoltán Szentmiklóssy²

¹ Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences 13–15 Reáltanoda u. 1053 Budapest, Hungary
² Institute of Mathematics Faculty of Science Eötvös Loránd University Pázmány Péter sétány 1/C 1117 Budapest, Hungary

Fundamenta Mathematicae, Tome 228 (2015) no. 1, pp. 27-46.

Résumé

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.

Détail

DOI : 10.4064/fm228-1-3

Keywords: improve results pavlov filatova concerning problem malykhin showing every regular space satisfies vardelta operatorname omega resolvable here vardelta dispersion character smallest size non empty set operatorname extent supremum sizes closed and discrete subsets particular regular lindel spaces uncountable dispersion character omega resolvable prove regular lindel space vardelta omega even omega resolvable question whether regular lindel spaces uncountable dispersion character maximally resolvable remains wide