Uniform Eberlein compactifications of metrizable spaces (Q409664)

In this paper, the authors investigate uniform Eberlein compactifications of metrizable spaces.NEWLINENEWLINEThey start by giving a short proof that every metrizable space has a uniform Eberlein compactification, which is a strengthening of a result of [\textit{A. V. Arkhangel'skii}, Topological function spaces. Mathematics and its Applications, Soviet Series, 78. Dordrecht: Kluwer Academic Publishers (1992; Zbl 0758.46026)]. As is noted in the paper, this first result was also obtained independently by \textit{B. A. Pasynkov} [Topology Appl. 159, No. 7, 1750--1760 (2012; Zbl 1252.54020)]; although Pasynkov's result yields such a compactification with stronger properties, the proof presented here is more straightforward. The authors also show that, if (and only if) the metrizable space has size not exceeding the continuum, one can obtain a first countable such compactification.NEWLINENEWLINEThe last two thirds of the paper are dedicated to the study of scattered uniform Eberlein compactifications of scattered metrizable spaces. The authors characterize the class of compact scattered hereditarily paracompact Hausdorff spaces as the smallest class \(\mathcal A\) that contains the finite discrete spaces and is closed under taking the Alexandroff compactifications of topological sums of elements of \(\mathcal A\). It is then proved that every space in the class \(\mathcal A\) is a uniform Eberlein compact, and that every scattered metrizable space has a compactification that belongs to \(\mathcal A\).