The partially ordered set of one-point extensions

TL;DR

This paper studies the structure of one-point extensions of Tychonoff spaces, establishing a correspondence with compact subsets of the outgrowth and proposing a conjecture relating order-isomorphism of zero-sets to homeomorphism of certain subspaces.

Contribution

It introduces an anti-order-isomorphism between one-point extensions and compact subsets of the outgrowth, enabling analysis of their order structure through topological properties.

Findings

Established a correspondence between extensions and subsets of the outgrowth.

Connected the order structure of extensions to topological features of the outgrowth.

Proposed a conjecture relating order-isomorphism of zero-sets to homeomorphism of specific subspaces.

Abstract

A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\em one-point extension} of $X$ if $Y\backslash X$ is a singleton. Let ${\mathcal P}$ be a topological property. An extension $Y$ of $X$ is called a {\em ${\mathcal P}$-extension} of $X$ if it has ${\mathcal P}$. One-point ${\mathcal P}$-extensions comprise the subject matter of this article. Here ${\mathcal P}$ is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space $X$ (partially ordered by $\leq$) and the set of compact non-empty subsets of its outgrowth $\beta X\backslash X$ (partially ordered by $\subseteq$). This enables us to study the order-structure of various sets of one-point extensions of the space $X$ by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces $X$ denote by ${\mathscr U}(X)$ the set of all zero-sets of $\beta X$ which miss $X$. \noindent{\bf Conjecture.} {\em For locally compact spaces $X$ and $Y$ the partially ordered sets $({\mathscr U}(X),\subseteq)$ and $({\mathscr U}(Y),\subseteq)$ are order-isomorphic if and only if the spaces ${\em cl}_{\beta X}(\beta X\backslash\upsilon X)$ and ${\em cl}_{\beta Y}(\beta Y\backslash\upsilon Y)$ are homeomorphic.}