What is 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 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 X ∖ { p } {\displaystyle X\setminus \{p\}} has the discrete topology, then the open extension topology on ∪ { p } {\displaystyle \cup \{p\}} is the excluded point topology.

This topology is used to provide interesting examples and counterexamples.