Completions and compactifications of quasi-uniform spaces

A ∗-compactification of a T1 quasi-uniform space (X,U) is a compact T1 quasi-uniform space (Y,V) that has a T(V∗)-dense subspace quasi-isomorphic to (X,U), where V∗ denotes the coarsest uniformity finer than V. he help of the notion of T1 ∗-half completion of a quasi-uniform space, which is introduced and studied here, we show that if a T1 quasi-uniform space (X,U) has a ∗-compactification, then it is unique up to quasi-isomorphism. We identify the ∗-compactification of (X,U) with the subspace of its bicompletion (X,U) consisting of all points which are closed in (X,T(U)) and prove that (X,U) is ∗-compactifiable if and only if it is point symmetric and (X,U) is compact. Finally, we discuss some properties of locally fitting T0 quasi-uniform spaces, a large class of quasi-uniform spaces whose bicompletion is T1, and, hence, they are T1 ∗-half completable.