Separation properties of the Wallman ordered compactification

Abstract

The Wallman ordered compactification ω0X of a topological ordered space X is T2-ordered (and hence equivalent to the Stone-Čech ordered compactification) iff X is a T4-ordered c-space. In particular, these two ordered compactifications are equivalent when X is n dimensional Euclidean space iff n≤2. When X is a c-space, ω0X is T1-ordered; we give conditions on X under which the converse statement is also true. We also find conditions on X which are necessary and sufficient for ω0X to be T2. Several examples provide further insight into the separation properties of ω0X.

Keywords

• c-set
• maximal c-filter
• T1-ordered space
• T2-ordered space
• ordered compactification.