Vector metric spaces and some properties (Q981946)

Let \(X\) be a nonempty set, \(E\) a Riesz space and \(d: X\times X\to E\) such that \(d(x,y)= 0\) if and only if \(x=y\) and \(d(x,y)\leq d(x,z)+ d(z,y)\) for all \(x,y,z\in X\). The triple \((X,d,E)\) is called a vector metric space. The authors study the basic properties of these spaces and prove that if \(X\) is \(E\)-complete and \(E\) is Archimedean then the Banach fixed point theorem holds. If moreover \(E\) is Dedekind complete they also prove the Baire theorem.