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 $f:Y_1\rightarrow Y_2$ 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 $Y_2\leq Y_1$, if there exists a continuous function $f:Y_1\rightarrow Y_2$ which keeps $X$ pointwise fixed. This is a partial order on the set of all Tychonoff extensions of $X$. If a Tychonoff extension $Y$ of $X$ is such that $Y\backslash X$ 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 $X$. We study the order structure of the partially ordered set $(T(X),\leq)$. For a locally compact space $X$, we define an order-anti-isomorphism from $T(X)$ onto the set of all non-empty closed subsets of $\beta X\backslash X$. 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 Lindelof extensions of $X$, the set of all one-point pseudocompact extensions of $X$, and the set of all one-point Cech-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 $\beta X\backslash X$. 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 $C^*(X)$ and the topology of subspaces of $\beta X\backslash X$.