On Borel almost disjoint families (Q691029)

To every uncountable almost disjoint family \( \mathcal{A} \) of infinite subsets of a countable set \( D \), one can canonically associate an Aleksandrov-Urysohn compactum (briefly: \( \mathsf{AU} \)-compactum) \( K_{\mathcal{A}} \), i.e.\ an uncountable separable compact space whose set of accumulation points has exactly one non-isolated point. Conversely, from an \( \mathsf{AU} \)-compactum \( K \) and a given countable dense \(D \subseteq K\) one can canonically extract an uncountable almost disjoint family \( \mathcal{A} \) of infinite subsets of \( D \) such that \(K\) and \(K_{\mathcal{A}}\) are homeomorphic (cf.\ pp.\ 547--548). In this nice paper, the authors relate the topological complexity of a family \( \mathcal{A} \) as above to the properties of \( K_{\mathcal{A}} \) and of the associated Banach space \( C(K_{\mathcal{A}}) \) of real-valued continuous functions on \( K_{\mathcal{A}} \). The main result is the following (Theorem 1.1): \noindent For any \(\mathsf{AU}\)-compactum \(K\) the following conditions are equivalent: \noindent (i) \(K\) embeds into the space \(B_1(2^\omega)\) of bounded real-valued first Baire class functions on the Cantor space \(2^\omega\) (equipped with the pointwise topology); \noindent (ii) the space \((C(K), Cyl(C(K)))\), where \(Cyl(C(K))\) is the cylindrical \(\sigma\)-algebra on \(C(K)\), i.e.\ the smallest \(\sigma\)-algebra for which all functionals in the dual \((C(K))^*\) are measurable, is standard Borel (equivalently: for any countable dense \(D \subseteq K\), the space \(C_D(K) = \{ f \restriction D \mid f \in C(K) \}\) is a Borel subset of \(\mathbb{R}^D\)); \noindent (iii) \(K\) is of the form \(K_{\mathcal{A}}\) for some Borel uncountable almost disjoint family \(\mathcal{A}\). Actually, the authors show that the implication (i) \(\Rightarrow\) (ii) holds for arbitrary separable compacta, and that \(\mathsf{AU}\)-compacta satisfying (i) can in fact be embedded in the unit ball of the double dual of a separable Banach space not containing any copy of \(\ell_1\) (equipped with the weak\({}^*\) topology). Moreover, a counterpart of the main theorem concerning the Johnson-Lindenstrauss spaces \(\mathsf{JL}(K)\) canonically associated to the \(\mathsf{AU}\)-compacta \(K\) is presented in Theorem 9.1. Finally, the authors observe that there are continuum many topological types of \(\mathsf{AU}\)-compacta \(K_{\mathcal{A}}\) with \(\mathcal{A} \subseteq 2^\omega\) compact (Section 7), and that there is a sequence \((\mathcal{A}_\alpha)_{\alpha < \omega_1}\) of Borel almost disjoint families such that \((K_{\mathcal{A}_\alpha})_{\alpha < \omega_1}\) is cofinal among the \(\mathsf{AU}\)-compacta considered in the main theorem: for every Borel almost disjoint family \(\mathcal{A}\) there is \( \alpha< \omega_1\) such that \(K_{\mathcal{A}}\) embeds (as a retract) into \(K_{\mathcal{A}_\alpha}\) (Theorem 1.2).