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.
This topology is used to provide interesting examples and counterexamples.
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.