The partially ordered set of one-point extensions

摘要

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~1 of X let Y ≤ Y~1 if there is a continuous function of Y~1 into Y which fixes X point-wise. An extension Yof X is called a one-point extension of X if Y X is a singleton. Let Ρ be a topological property. An extension Y of X is called a V-extension of X if it has Ρ. One-point Ρ-extensions comprise the subject matter of this article. Here Ρ 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 βXX (partially ordered by (c)). 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),(c)) and ((U)(Y), c) are order-isomorphic if and only if the spaces cl_(βx)(βX υX) and cl_(βY)(βY υY) are homeomorphic.