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

${\displaystyle T=\{S\subseteq X\mid p\in S{\text{ or }}S=\emptyset \}}$

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.