Particular point topology

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 set and "p" ∈ "X". The collection
of subsets of "X" is the particular point topology on "X". There are a variety of cases which 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.