Particular point topology

rdf:langString Particular point topology

rdf:langString In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is the particular point topology on X. There are a variety of cases that are individually named: * If X has two points, the particular point topology on X is the Sierpiński space. * If X is finite (with at least 3 points), the topology on X is called the finite particular point topology. * If X is countably infinite, the topology on X is called the countable particular point topology. * If X is uncountable, the topology on X is called the uncountable particular point topology. A generalization of the particular point topology is the closed extension topology. In the case when X \ {p} has the discrete topology, the closed extension topology is the same as the particular point topology. This topology is used to provide interesting examples and counterexamples.