The partially ordered set of one-point extensions

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• 作者：M.R. Koushesh^a ; ^b ; ^{koushesh@cc.iut.ac.ir}
• 关键词：Stone– ; Č ; ech compactification ; Hewitt realcompactification ; One-point extension ; One-point compactification ; Mró ; wka's condition (W) ; Mró ; wka topological property ; Axiom Ω ;
• 刊名：Topology and its Applications
• 出版年：2011
• 出版时间：15 February 2011
• 年：2011
• 卷：158
• 期：3
• 页码：509-532
• 全文大小：323 K

文摘

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 YY^′ 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 YX is a singleton. Let be a topological property. An extension Y of X is called a -extension of X if it has .

One-point -extensions comprise the subject matter of this article. Here 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 ). 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 the set of all zero-sets of βX which miss X.

Conjecture

For locally compact spaces X and Y the partially ordered sets and are order-isomorphic if and only if the spaces cl_βX(βXυX) and cl_βY(βYυY) are homeomorphic.