A new characterization of Eberlein compacta (Q2717582)

It is known that every Eberlein compact space is Radon-Nikodým compact. A necessary and sufficient condition for a Radon-Nikodým compact space to be Eberlein compact is that it is Corson compact [\textit{J. Orihuela, W. Schachermayer} and \textit{M. Valdivia}, Stud. Math. 98, No. 2, 157-174 (1991; Zbl 0771.46015); see also \textit{C. Stegall}, Acta Univ. Carol., Math. Phys. 32, No. 2, 47-54 (1991; Zbl 0773.46008)]. NEWLINENEWLINENEWLINEThe main result of this paper is a characterization of Eberlein compact spaces. For that purpose, the author introduces the following two notions: NEWLINENEWLINENEWLINE1) a set \(X\) is said to have property \({\mathcal L}(\tau_1,\tau_2)\), where \(\tau_1\) and \(\tau_2\) are topologies on \(X\), if, for any \(x\in X\), there is a countable set \(S(x)\), containing \(x\), such that \({\overline A}^{\tau_2}\subseteq \overline{\bigcup\{S(x) ;\;x\in A\}}^{\tau_1}\) for any \(A\subseteq X\); NEWLINENEWLINENEWLINE2) a topological space \((X,\tau)\) is said to have the linking separability property (LSP) if there exists a metric \(d\) on \(X\) whose topology is finer than \(\tau\) such that \(X\) has \({\mathcal L}(d,\tau)\).NEWLINENEWLINENEWLINEThe author proves that if \((K,\tau)\) is a compact Hausdorff space so that: \((\ast)\) There exists a lower semi-continuous metric \(\rho\) on \(K\) such that \((K,\tau)\) has \({\mathcal L}(\rho,\tau)\), then \(K\) is Corson compact (Theorem 1.6) and \(K\) is Radon-Nikodým compact (Proposition 1.7). This last fact follows from \textit{I. Namioka}'s characterization [Mathematika 34, No. 2, 258-281 (1987; Zbl 0654.46017)]: a compact Hausdorff space is Radon-Nikodým compact if and only if it is fragmentated by a l.s.c. metric. For the first result, the author constructs for every compact Hausdorff space \((K,\tau)\) with property \((\ast)\) a projectional resolution of the identity \((P_\alpha)\) in \({\mathcal C}(K)\) and a family of continuous retractions \(r_\alpha\colon K\to K\) such that \(P_\alpha(f)=f\circ r_\alpha\) (Theorem 1.4) and then argues by induction on the density character of \(K\), beginning with the case where \((K,\tau)\) is separable (which is proved in Proposition 1.7). Using next a result of his thesis (any WCG Banach space has \({\mathcal L}(\|\;\|,\text{weak})\)), he is able to conclude that a compact Hausdorff space \((K,\tau)\) is Eberlein compact if and only if it has \((\ast)\) (Theorem A) (note that in the proof of Theorem A, the implications (i) \(\Rightarrow\) (ii) and (ii) \(\Rightarrow\) (i) should be reversed). That allows him to deduce that a Radon-Nikodým compact space is Eberlein compact if and only if it has the LSP (Theorem B).NEWLINENEWLINENEWLINEThe paper is ended with examples showing that, for \((X,\tau)\) topological space, the properties: \((X,\tau)\) has \({\mathcal L}(\rho,\tau)\), \((X,\tau)\) is \(\rho\)-\(\sigma\)-fragmentated, and another property, \((X,\tau)\) has \(\rho\)-SLD (\(\rho\) metric on \(X\) with topology finer than \(\tau\)), which are equivalent for metrizable \((X,\tau)\), are in general distinct.