Spaces of continuous functions defined on Mrówka spaces
A collection \({\mathcal A}\) of subsets of the natural numbers \(\omega\) is an almost disjoint family if each \(A\) in \({\mathcal A}\) is infinite, and for each two different elements \(A\), \(B\) in \({\mathcal A}\), \(|A\cap B|<\aleph_0\). A maximal almost disjoint family is a maximal element in the family of all almost disjoint families with the containment order. A topological space \(\Psi({\mathcal A})\) is a Mrówka space if it has the form \(\omega\cup{\mathcal A}\), where \({\mathcal A}\) is an almost disjoint family, and its topology is generated by the following base: each \(\{n\}\) is open for every \(n\in\omega\), and an open canonical neighborhood of \(A\in{\mathcal A}\) is of the form \(\{{\mathcal A}\}\cup B\) where \(B\subset\omega\) and \(A\setminus B\) is finite. The authors discuss normality and the Lindelöf property in spaces of continuous functions over \(\Psi({\mathcal A})\) and obtain the following results. Theorem 2.7. Let \(E\in\{\mathbb{I},\mathbb{R},\mathbb{P},2^\omega\}\). For every uncountable almost disjoint family \({\mathcal A}\) and every compactification \(b\Psi(A)\) of \(\Psi({\mathcal A})\), the space \(C_p(b\Psi({\mathcal A}),E)\) is not normal, where \(\mathbb{R}\) and \(\mathbb{P}\) are the spaces of real and irrational numbers with the natural topology, respectively, and \(\mathbb{I}\) denotes the unit interval \([0,1]\subset\mathbb{R}\). Theorem 3.4. Let \({\mathcal A}\) be an infinite maximal almost disjoint family on \(\omega\). Then the spaces \(C_p(\Psi({\mathcal A}), 2^\omega)\), \(C_p(\Psi({\mathcal A}), \omega^\omega)\), \(C_p(\Psi({\mathcal A}),\mathbb{I})\) and \(C_p(\Psi({\mathcal A}),\mathbb{R})\) are not normal, and their extent and Lindelöf numbers are all equal to \(|{\mathcal A}|\), where the extent of \(X\) means the supremum of the cardinalities of all the closed and discrete subspaces of \(X\). An almost disjoint family \({\mathcal A}\) of subsets of \(\omega\) is quasi-maximal if there is a maximal almost disjoint family \({\mathcal B}\) such that \(|{\mathcal B}\setminus{\mathcal A}|\leq\aleph_0\). The authors show the above theorems to remain true if quasi-maximal almost disjoint families replace maximal almost disjoint families. Theorem 4.5. Assume CH. There is a Mrówka maximal almost disjoint family \({\mathcal A}\) such that \(C_p(\Psi({\mathcal A},2))\) is Lindelöf.
function space with the pointwise convergence topology
scientific article; zbMATH DE number 2142293
scientific article