In this paper we introduce new characterizations of the Wallman compactification among theT1compactifications of a space. We then use one of these characterizations to investigate conditions which would imply that an intermediate Wallman space have a Wallman compactification homeomorphic to the original compactification. The most general of these conditions is that disjoint closed subsets of the intermediate space have disjoint closures in the compactification. This has, as a special case, the consequence that ifX ⊆ Y ⊆ Xand ifY\Xis closed inWX\X, thenWY≅W X.