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 formula_2 has the discrete topology, then the open extension topology on formula_3 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.