A compactification of locally compact spaces

Abstract:

Every locally compact space $X$ has its topology determined by its 1-1 compact images and hence has a compactification $\xi X$ obtained as the closure of the natural embedding of $X$ in the product of those images, just as the Stone-Čech compactification $\beta X$ can be obtained by embedding $X$ in a product of intervals. The obvious question is whether $\xi X = \beta X$. In this paper we prove that $\xi X = \beta X$ if $X$ either is $0$-dimensional or contains an arc, and give an example in which $\xi X \ne \beta X$.