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 $\mathcal P \left({S}\right)$ as:

$\tau_p = \left\{{A \subseteq S: p \in A}\right\} \cup \left\{{\varnothing}\right\}$

... 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 $T = \left({S, \tau_p}\right)$ is called the particular point space on $S$ by $p$, or just a particular point space.

Let $S$ be finite.

Then $\tau_p$ is a finite particular point topology, and $\left({S, \tau_p}\right)$ is a finite particular point space.

Let $S$ be infinite.

Then $\tau_p$ is an infinite particular point topology, and $\left({S, \tau_p}\right)$ is an infinite particular point space.