# Definition:Particular Point Topology

## Contents

## 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**.

### Finite Particular Point Topology

Let $S$ be finite.

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

### Infinite Particular Point Topology

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**.

## Also see

- Results about
**particular point topologies**can be found here.

## Sources

- 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