Topological extensions with compact remainder

TL;DR

This paper explores the relationship between the order structure of certain topological extensions of a space with compact remainder and the topology of subspaces of its Stone-Čech compactification, focusing on pseudocompactness and realcompactness.

Contribution

It establishes connections between the order structure of $rak{P}$-extensions and the topology of subspaces of $eta X ackslash X$, specifically for pseudocompactness and realcompactness.

Findings

Characterizes the order structure of $rak{P}$-extensions with compact remainder.

Analyzes the topology of subspaces of $eta X ackslash X$ related to these extensions.

Provides detailed results for pseudocompactness and realcompactness cases.

Abstract

Let $\mathfrak{P}$ be a topological property. We study the relation between the order structure of the set of all $\mathfrak{P}$-extensions of a completely regular space $X$ with compact remainder (partially ordered by the standard partial order $\leq$) and the topology of certain subspaces of the outgrowth $\beta X\setminus X$. The cases when $\mathfrak{P}$ is either pseudocompactness or realcompactness are studied in more detail.