# Definition:Particular Point Topology/Infinite

## Definition

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

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

Let $T = \left({S, \tau_p}\right)$ be the particular point space on $S$ by $p$.

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.

### Countable Particular Point Topology

Let $S$ be countably infinite.

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

### Uncountable Particular Point Topology

Let $S$ be uncountable.

Then $\tau_p$ is an uncountable particular point topology, and $\struct {S, \tau_p}$ is an uncountable particular point space.

## Also see

• Results about particular point topologies can be found here.