On Eberlein compactifications of metrizable spaces (Q2773368)

Every metrizable space has an Eberlein compactification \(eX\), cf. \textit{A.V. Arkhangel'skij}, [Topological Function Spaces, Mathematics and its Applications, Soviet Series. 78. Kluwer (1992; Zbl 0758.46026)]. It is shown here that one can find \(eX\) such that \(w(X)=w(eX)\) and dim\(\,X= \) dim\(\, eX\) (if dim\(\,X<\infty\)) or both \(X\) and \(eX\) are S-weakly infinite-dimensional. The Čech-Stone compactification \(\beta X\) is a supremum of all Eberlein compactifications of \(X\) provided \(X\) has an Eberlein compactification.