Uniform Eberlein compactifications of metrizable spaces

We prove that each metrizable space X (of size |X|⩽c) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class A of compact spaces, which contains the empty set, the singleton, and is closed under producing the Alexandroff compactification of the topological sum of a family of compacta from the class A.