Separated and prime compactifications

TL;DR

This paper explores how various prime and separated compactifications of topological spaces can be characterized categorically through ultrafilter space monads and reflectors, unifying multiple constructions within a common framework.

Contribution

It introduces a categorical framework that encompasses several prime and separated compactification constructions using ultrafilter space monads and reflectors.

Findings

Prime open filter monad fits into the categorical framework.

Prime closed filter compactification is unified under the same framework.

Separated completion monad is also included in the categorical approach.

Abstract

We discuss conditions under which certain compactifications of topological spaces can be obtained by composing the ultrafilter space monad with suitable reflectors. In particular, we show that these compactifications inherit their categorical properties from the ultrafilter space monad. We further observe that various constructions such as the prime open filter monad defined by H. Simmons, the prime closed filter compactification studied by Bentley and Herrlich, as well as the separated completion monad studied by Salbany fall within the same categorical framework.