The partially ordered set of one-point extensions

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 $Y^{\mathrm{\prime }}$ of $X$ let $Y\le Y^{\mathrm{\prime }}$ if there is a continuous function of $Y^{\mathrm{\prime }}$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\em one-point extension} of $X$ if $Y\mathrm{\backslash }X$ 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 $\le$) and the set of compact non-empty subsets of its outgrowth $\beta X\mathrm{\backslash }X$ (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 $\mathcal{U}(X)$ 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 $(\mathcal{U}(X),\subseteq )$ and $({\mathscr U}(Y),\subseteq)$areorder-isomorphicifandonlyifthespaces${\em cl}{\beta X}(\beta X\backslash\upsilon X)$and${\em cl}{\beta Y}(\beta Y\backslash\upsilon Y)$ are homeomorphic.}