Key properties -- what are the key properties of partial metric spaces?Let (X,p) be a partial metric space. Then the sets Be(x) = { y | p(x,y) < e }, termed open balls, form a basis for a T0 topology [Mat94].Let (X,p) be a partial metric space. Let d(x,y) = 2×p(x,y) - p(x,x) - p(y,y). Then (X,d) is a metric space. The usual open ball toplogy of (X,d) is a sub topology of the T0 topology of (X,p) [Mat94].
Let (X,p) be a partial metric space. Let q(x,y) = p(x,y) - p(x,x). Then (X,q) is a quasi-metric space [Mat94]. Note that p(x,y)=q(x,y)+q(y,x). Suppose d:X×X->R+ and | |:X->R+ are functions such that d(x,y) >= |x|-|y| for all x and y in X. Then (X,d,| |) is termed a weighted metric space. Let p(x,y) = (d(x,y)+|x|+|y|)/2. Then (X,p) is a partial metric space [Mat94]. Note that p(x,x) = |x|.
Suppose q:X×X->R+ and | |:X->R+ are functions such that q(x,y)+|x| = q(y,x)+|y| for all x and y in X. Then (X,q) is termed a weighted quasi-metric space. Let p(x,y) = q(x,y)+|x|. Then (X,p) is a partial metric space [Mat94]. Note that p(x,x) = |x|. The Banach contraction mapping theorem [Su75] for complete metric spaces naturally extends to complete partial metric spaces [Mat94]. |