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.

Let $S$ be finite.

Then $\tau_p$ is a finite particular point topology, and $\struct {S, \tau_p}$ 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.

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology