Definition:Particular Point Topology

Definition
Let $S$ be a set which is non-empty.

Let $p \in S$ be some particular point of $S$.

We define a subset $\tau_p$ of the power set $\powerset S$ as:
 * $\tau_p = \set {A \subseteq S: p \in A} \cup \set \O$

that is, all the subsets of $S$ which include $p$, along with the empty set.

Then $\tau_p$ is a topology called the particular point topology on $S$ by $p$, or just a particular point topology.

The topological space $T = \struct {S, \tau_p}$ is called the particular point space on $S$ by $p$, or just a particular point space.