  A semigroup A is an abelian semigroup with identity 0. A set of positives in
A is an ordered down-directed set P containing with every r an element r/2 with
r/2 + r/2 = r. A continuity space is an abstract set X equipped with a map d :
XxX to A satisfying d(x, x) = 0 and d(x, z) d(x, y) + d(y, z). A quasi-uniform
space is an abstract set X equipped with a filterbase of binary relations {U}
such that each U contains the diagonal as well as for some V{U}. For each rP,
the set } is seen to be a quasi-uniform filterbase on X . Indeed, the
down-directedness of P ensures that U(r) is a filterbase of oversets of the
diagonal and U(r) contains U(r/2)U(r/2). One obtains a uniform filterbase by
symmetrization, i.e. by intersecting the U(r) with the U(s) = {(y, x)|d(y, x)
<s}.
