The partially ordered set of one-point extensions

Abstract

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 β X ∖ 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 cl β X ( β X ∖ υ X ) and cl β Y ( β Y ∖ υ Y ) are homeomorphic.