Uniform Eberlein compactifications of metrizable spaces

TL;DR

This paper demonstrates that all metrizable spaces of size up to continuum have a uniform Eberlein compactification, and characterizes scattered metrizable spaces with scattered hereditarily paracompact compactifications, expanding understanding of their structure.

Contribution

It establishes the existence of uniform Eberlein compactifications for metrizable spaces and characterizes scattered hereditarily paracompact compact spaces within a specific class.

Findings

Every metrizable space of size ≤ continuum has a uniform Eberlein compactification.

Scattered metrizable spaces have scattered hereditarily paracompact compactifications.

The class of compact scattered hereditarily paracompact spaces is minimal and closed under specific operations.

Abstract

We prove that each metrizable space (of cardinality less or equal to continuum) 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 of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class.