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: 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.

Properties

Connectedness Properties

For any x, y ∈ X, the function f: [0, 1] → X given by is a path. However, since p is open, the preimage of p under a continuous injection from [0,1] would be an open single point of [0,1], which is a contradiction.

Compactness Properties

Limit related

Separation related

Other properties