Excluded point topology


In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any 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. A space with the excluded point topology is connected, since the only open set containing the excluded point is X itself and hence X cannot be written as disjoint union of two proper open subsets.