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Large open covers and a polarized partition relation. (English) Zbl 0903.03028

First some definitions. Let \(X\) be a fixed topological space. An open cover \(\mathcal U\) of \(X\) is an \(\omega\)-cover if \(X\) is not an element of \(\mathcal U\) and every finite subset of \(X\) is contained in an element of \(\mathcal U\). We write \(\Omega\) for the collection of all open covers of \(X\). We say \(X\) is an \(\varepsilon\)-space if every \(\omega\)-cover of \(X\) has a countable subset which is an \(\omega\)-cover of \(X\). \(X\) has property \(\mathbf{S}_1(\Omega,\Omega)\) if for every sequence \(({\mathcal U}_n : n=1,2,3,\dots)\) of \(\omega\)-covers of \(X\) there is a sequence \((U_n : n=1,2,3,\dots)\) such that \(U_n\in{\mathcal U}_n\) for each \(n\), and \(\{U_n : n=1,2,3,\dots\}\) is an \(\omega\)-cover of \(X\). \(X\) has property \(\mathbf{S}_{\text{fin}}(\Omega,\Omega)\) if for every sequence \(({\mathcal U}_n : n=1,2,3,\dots)\) of \(\omega\)-covers of \(X\) there is a sequence \(({\mathcal V}_n : n=1,2,3,\dots)\) such that \({\mathcal V}_n\) is a finite subset of \({\mathcal U}_n\) for each \(n\), and \(\bigcup_{n=1}^{\infty}{\mathcal V}_n\) is an \(\omega\)-cover of \(X\). And \(X\) has the property \(\mathbf{Split}(\Omega,\Omega)\) if for each \(\omega\)-cover \(\mathcal U\) of \(X\), there are two disjoint \(\omega\)-covers \({\mathcal B}_1\) and \({\mathcal B}_2\) such that \({\mathcal B}_1 \cup {\mathcal B}_2 \subseteq {\mathcal U}\).
The paper is concerned with the following weak square bracket polarized partition property, which for this review we shall call property \((\ast)\): For every \({\mathcal A}_1,{\mathcal A}_2 \in \Omega\) and every colouring \(f:{\mathcal A}_1\times {\mathcal A}_2 \to \{1,2,\dots,k\}\) of \({\mathcal A}_1\times {\mathcal A}_2\) by a finite number \(k\) of colours, there are \({\mathcal B}_1,{\mathcal B}_2 \in \Omega\) such that \({\mathcal B}_1 \subseteq {\mathcal A}_1\), \({\mathcal B}_2 \subseteq {\mathcal A}_2\) and \({\mathcal B}_1\times {\mathcal B}_2\) takes only two colours (i.e. \(| \{f(x,y) : (x,y) \in {\mathcal B}_1\times {\mathcal B}_2\}| \leq 2\)).
The author proves: If \(X\) has property \(\mathbf{S}_1(\Omega,\Omega)\) then \(X\) has property \((\ast)\). If \(X\) is an \(\varepsilon\)-space with property \((\ast)\) then \(X\) has the two properties \(\mathbf{S}_{\text{fin}} (\Omega,\Omega)\) and \(\mathbf{Split}(\Omega,\Omega)\).

MSC:

03E05 Other combinatorial set theory
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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References:

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