Particular Point Topology (Paperback)

High Quality Content by WIKIPEDIA articles! The particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. 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. Since every nonempty open set contains p, no nonempty closed sets can be separated by neighbourhoods from each other, and thus X is not normal. Exception: the Sierpinski topology is normal, and even completely normal, since it contains no nontrivial separated sets.