Examples -- what are some examples of partial metric spaces?

1. Let R⁺ denote the set of all non negative real numbers. Let p:R⁺×R⁺->R⁺ be such that p(x,y) = x if y <= x, and y otherwise. Then (R⁺,p) is a partial metric space. Thus p(x,y) is defined to be the maximum of x and y. Then (R⁺,p) is a partial metric space. The usual ordering x <= y if and only if p(x,x) = p(x,y) for partial metric spaces is thus x greater than or equal to y.
   
   
2. Let I denote the set of all intervals [a,b] for any real numbers a <= b. Let p:I×I->R⁺ be the function such that p([a,b], [c,d]) = max(b,d) - min(a,c). Then (I,p) is a partial metric space such that [a,b] <= [c,d] if and only if [c,d] is contained within [a,b]. The self distance p([a,b],[a,b]) is b-a, the length of the interval [a,b].
   
   
3. In Scott's theory of domains [AJ94] the so-called flat domains are introduced. A flat domain is a set X with an element _|_ (not in X) partially ordered by, x <= y if and only if x = _|_ or x = y. Thus _|_ (pronounced bottom) is the least member of the ordering. Let p be the function such that p(x,y) = 0 if x=y is in X, and 1 otherwise. Then (X,p) is a partial metric,space, and its ordering is precisely the ordering of a flat domain.
   
   
4. Let (X,d) be a metric space. Let FB be the set of all formal balls, that is, pairs (x,r) such that x is in X and r is a non negative real number. Formal balls of a complete metric space are of interest in domain theory as they form a so-called domain representation for that space. Let p((x,r),(y,s)) = max{r, (d(x,y)+r+s)/2, s}. Then it can be shown that p is a partial metric such that the ordering (x,r) <= (y,s) iff p((x,r),(x,r)) = p((x,r),(y,s)) is equivalent to the usual ordering (x,r) <= (y,s) iff d(x,y) <= r-s on formal balls.