Dislocated Topologies

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We study a generalized notion of topology which evolved out of 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. The corresponding generalized notion of metric is obtained by allowing points to have non-zero distance to themselves. We further show that it is meaningful to discuss neighbourhoods, convergence, and continuity in these spaces. A generalized version of the Banach contracting mapping theorem can also be established. We show finally how the generalized metrics studied here can be obtained from conventional metrics.