  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.}
