# Hausdorffness of General Compactifications

## TL;DR

This paper characterizes partitions in Stone-Čech compactifications that lead to Hausdorff compactifications, and constructs embeddings of certain lattices of compactifications, advancing understanding of their structural properties.

## Contribution

It provides explicit characterizations of partitions leading to Hausdorff compactifications and constructs embeddings of upper semi-lattices into lattices of compactifications.

## Key findings

- Partitions in Stone-Čech compactifications are characterized for Hausdorff compactifications
- Embeddings of upper semi-lattices into lattices of compactifications are constructed
- The relationship between lattice isomorphisms and homeomorphic remainders is clarified

## Abstract

Magill proved that the remainders of two locally compact Hausdorff spaces in their StoneCech compactifications are homeomorphic if and only if the lattices of their Hausdorff compactifications are lattice isomorphic. His construction for compactifications are explicitely discussed through the partitions of their StoneCech compactifications. Partitions in a StoneCech compactification which lead to Hausdorff compactifications are characterized in this article. Embeddings of certain upper semi-lattices of compactifications into lattices of compactifications are constructed.