ABSTRACT
On the other hand, working with nonsymmetric distances has some advantages with respect to symmetric distances. For example, we recall that in a computational process one can define a partial order given by x < y if and only if y contains more information than x. In many cases this fact can be interpreted by means of an appropriate To topology, where x < y if and only if x G {y}. When we consider a metrizable topological space (X, r) then it is Hausdorff, so x ^ y whenever x ^ y, hence only equals elements are related. This is due to the facts that d(x,y) = d(y,x), and d(x,y) = 0 implies x — y. Therefore, if we consider a quasi-metric on X (see [9]) which is a function d : X x X ^ R + such that i) d(x,y) = d(y,x) = 0 if and only if x = y\ and ii) d(x,z) < d(x}y) + d(y,z), for all x,y,z G X; then we avoid the above problems. A quasi-metric is always compatible with the specialization order, in the following sense x < y if and only if d(x,y) = 0. This is useful because if d(x,y) = 0 and d(y,x) > 0 for two different points then we know that x < y. Consequently, when obtaining the distance between two points we not only obtain the "difference" between that points but also an order relation.
