In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is then the excluded point topology on X. There are a variety of cases which are individually named: A generalization is the open extension topology; if has the discrete topology, then the open extension topology on is the excluded point topology. This topology is used to provide interesting examples and counterexamples.

Properties

Let X be a space with the excluded point topology with special point p. The space is compact, as the only neighborhood of p is the whole space. The topology is an Alexandrov topology. The smallest neighborhood of p is the whole space X; the smallest neighborhood of a point x\ne p is the singleton {x}. These smallest neighborhoods are compact. Their closures are respectively X and {x,p}, which are also compact. So the space is locally relatively compact (each point admits a local base of relatively compact neighborhoods) and locally compact in the sense that each point has a local base of compact neighborhoods. But points x\ne p do not admit a local base of closed compact neighborhoods. The space is ultraconnected, as any nonempty closed set contains the point p. Therefore the space is also connected and path-connected.