
arXiv: 1702.02454
Let us denote by $��(��,��)$ the statement that $\mathbb{B}(��) = D(��)^��$, i.e. the Baire space of weight $��$, has a coloring with $��$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(��)$ picks up all the $��$ colors. We call a space $X\,$ {\em $��$-regular} if it is Hausdorff and for every non-empty open set $U$ in $X$ there is a non-empty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called {\em feebly compact} if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact iff it is feebly compact. The main result of this paper is the following. Theorem. Let $X$ be a crowded feebly compact $��$-regular space and $��$ be a fixed (finite or infinite) cardinal. If $��(��,��)$ holds for all $��< \widehat{c}(X)$ then $X$ is $��$-resolvable, i.e. contains $��$ pairwise disjoint dense subsets. (Here $\widehat{c}(X)$ is the smallest cardinal $��$ such that $X$ does not contain $��$ many pairwise disjoint open sets.) This significantly improves earlier results of van Mill , resp. Ortiz-Castillo and Tomita.
8 pages
Consistency and independence results in general topology, Compactness, Cantor set, General Topology (math.GN), Baire space, feebly compact, pseudocompact, Metric spaces, metrizability, FOS: Mathematics, Cardinality properties (cardinal functions and inequalities, discrete subsets), resolvable, coloring, Mathematics - General Topology, 54D30, 54D99, 54A35
Consistency and independence results in general topology, Compactness, Cantor set, General Topology (math.GN), Baire space, feebly compact, pseudocompact, Metric spaces, metrizability, FOS: Mathematics, Cardinality properties (cardinal functions and inequalities, discrete subsets), resolvable, coloring, Mathematics - General Topology, 54D30, 54D99, 54A35
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