Abstract
The aim of this article is to introduce the notion of a (phi, psi) -metric space, which extends the metric space concept. In these spaces, the symmetry property is preserved. We present a natural topology tau((phi, psi)) in such spaces and discuss their topological properties. We also establish the Banach contraction principle in the context of (phi, psi)-metric spaces and we illustrate the significance of our main theorem by examples. Ultimately, as applications, the existence of a unique solution of Fredholm type integral equations in one and two dimensions is ensured and an example in support is given.