Differences -- how does a partial metric space differ from a metric space?A partial metric space is intended to be a smallest possible generalisation of the notion of metric space [Su75] such that the distance of each point from itself is no longer necessarily zero.Why would on earth would anyone want non zero self distance? For an answer please see The story. Thus the value p(x,x) may be either greater than or equal to zero. This value is termed the self-distance, size or weight of x. A metric space is thus precisely a partial metric space such that p(x,x) is always zero. Where appropriate size may be used to express the extent to which a point is partially defined. A partial metric space (X,p) can be partially ordered by the binary relation <= over S defined by x <= y if and only if p(x,x) = p(x,y). The partial ordering and size are related by the property, if x <= y then p(x,x) >= p(y,y). A metric space is precisely a partial metric space such that all sizes are zero, in which case the partial ordering is equality. |