Nonstandard Compactification of uniform spaces

Abstract

Let $(X, \Psi)$ be a uniform space. We define an equivalence relation on a superstructure ${ }^* X$ of $X$. The set of equivalence classes is denoted by $\bar{X}$. We extend the uniform structure $\Psi$ of $X$ to a suitable uniform structure $\widehat{\Psi}$ on $\bar{X}$. We embed $X$ as a dense subspace of $\bar{X}$ and show that $\bar{X}$ is compact. Thus $\bar{X}$ turns out to be a uniform compactification of $X$.