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.
- Hereditarily paracompact space
- Metrizable space
- Scattered compactification
- Scattered space
- Uniform Eberlein compact space
All Science Journal Classification (ASJC) codes
- Geometry and Topology