Compactifications are constructed for convergence ordered spaces and topological ordered spaces with extension properties that resemble those of the Stone-Čech compactification.

In this paper, we prove that a real valued bounded function, defined on a metric space and uniformly continuous is the uniform limit of a sequence of Lipschitzian bounded functions.

As a consequence, a new criterion for the weak convergence of probabilities is given.

Let B be a Douglas algebra which admits best approximation. It will be shown that the following are equivalent: (1) The unit ball of (L∞/B) has no extreme points; (2) For any Blaschke product b with , there exists h ∈ B such that = 1 and h|E≢0, where E is the essential set of B.

It will also be proven that if B⊇H∞+C and its essential set E contains a closed Gδ set, then the unit ball of (L∞/B) has no extreme points. Many known results concerning this subject will follow from these results.