Dislocated Topologies 1

We study a generalized notion of topology which evolved from applications in the area of logic programming semantics. The generalization is obtained by relaxing the requirement that a neighbourhood of a point includes the point itself, and by allowing neighbourhoods of points to be empty. We show that it is meaningful to discuss neighbourhoods, convergence, and continuity in these spaces. A generalized version of the Banach contraction mapping theorem can also be established. We will also show how the generalized metrics studied here can be obtained from conventional metrics.