# Article

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Keywords:
pseudocompact; feebly compact; resolvable; Baire space; coloring; Cantor set
Summary:
Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or infinite) cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. $X$ contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ is the smallest cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of [van Mill J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016].
References:
[1] Hajnal A., Juhász I., Shelah S.: Splitting strongly almost disjoint families. Trans. Amer. Math. Soc. 295 (1986), no. 1, 369–387. DOI 10.1090/S0002-9947-1986-0831204-9 | MR 0831204
[2] Hewitt E.: Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64 (1948), 45–99. DOI 10.1090/S0002-9947-1948-0026239-9 | MR 0026239
[3] Juhász I.: Cardinal Functions in Topology---Ten Years Later. Mathematical Centre Tracts, 123, Mathematisch Centrum, Amsterdam, 1980. MR 0576927
[4] Juhász I., Soukup L., Szentmiklóssy Z.: Resolvability of spaces having small spread or extent. Topology Appl. 154 (2007), no. 1, 144–154. DOI 10.1016/j.topol.2006.04.004 | MR 2271779
[5] Mardešić S., Papić P.: Sur les espaces dont toute transformation réelle continue est bornée. Hrvatsko Prirod. Društvo. Glasnik Mat.-Fiz. Astr. Ser. II. 10 (1955), 225–232 (French. Serbo-Croatian summary). MR 0080292
[6] van Mill J.: Every crowded pseudocompact ccc space is resolvable. Topology Appl. 213 (2016), 127–134. DOI 10.1016/j.topol.2016.08.020 | MR 3563074
[7] Ortiz-Castillo Y. F., Tomita A. H.: Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable. Conf. talk at Toposym 2016 available at http://www.toposym.cz//slides-Ortiz\_Castillo-2435.pdf
[8] Pavlov O.: Problems on (ir)resolvability. Open Problems in Topology. II. (Pearl E., ed.) Elsevier, Amsterdam, 2007, pages 51–59.
[9] Pytkeev E. G.: Resolvability of countably compact regular spaces. Proc. Steklov Inst. Math. 2002, Algebra. Topology. Mathematical Analysis, suppl. 2, S152–S154. MR 2068193
[10] Weiss W.: Partitioning topological spaces. Topology, Vol. II, Proc. Fourth Colloq., Budapest, 1978, Colloq. Math. Soc. János Bolyai, 23, North-Holland, 1980, pages 1249–1255. MR 0588871

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