The partially ordered set of one-point extensions

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be 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⩽Y′ if there is a continuous function of Y′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y∖X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here 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 ⩽) and the set of compact non-empty subsets of its outgrowth βX∖X (partially ordered by ⊆). 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 U(X) the set of all zero-sets of βX which miss X.Conjecture – For locally compact spaces X and Y the partially ordered sets (U(X),⊆) and (U(Y),⊆) are order-isomorphic if and only if the spaces clβX(βX∖υX) and clβY(βY∖υY) are homeomorphic.