On order structure of the set of one-point Tychonoff extensions of a locally compact space

Abstract

If a Tychonoff space X is dense in a Tychonoff space Y, then Y is called a Tychonoff extension of X. Two Tychonoff extensions Y 1 and Y 2 of X are said to be equivalent, if there exists a homeomorphism f : Y 1 → Y 2 which keeps X pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of X. We identify those extensions of X which belong to the same equivalence classes. For two Tychonoff extensions Y 1 and Y 2 of X, we write Y 2 ⩽ Y 1 , if there exists a continuous function f : Y 1 → Y 2 which keeps X pointwise fixed. This is a partial order on the set of all (equivalence classes of) Tychonoff extensions of X. If a Tychonoff extension Y of X is such that Y \\ X is a singleton, then Y is called a one-point extension of X. Let T ( X ) denote the set of all one-point extensions of X. Our purpose is to study the order structure of the partially ordered set ( T ( X ) , ⩽ ) . For a locally compact space X, we define an order-anti-isomorphism from T ( X ) onto the set of all nonempty closed subsets of β X \\ X . We consider various sets of one-point extensions, including the set of all one-point locally compact extensions of X, the set of all one-point Lindelöf extensions of X, the set of all one-point pseudocompact extensions of X, and the set of all one-point Čech-complete extensions of X, among others. We study how these sets of one-point extensions are related, and investigate the relation between their order structure, and the topology of subspaces of β X \\ X . We find some lower bounds for cardinalities of some of these sets of one-point extensions, and in a concluding section, we show how some of our results may be applied to obtain relations between the order structure of certain subfamilies of ideals of C ∗ ( X ) , partially ordered with inclusion, and the topology of subspaces of β X \\ X . We leave some problems open.