# Definition:Excluded 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 $\tau_{\bar p}$ of the power set $\mathcal P \left({S}\right)$ as:

- $\tau_{\bar p} = \left\{{A \subseteq S: p \notin A}\right\} \cup \left\{{S}\right\}$

That is, all the subsets of $S$ which do not include $p$, along with the set $S$.

Then $\tau_{\bar p}$ is a topology called the **excluded point topology on $S$ by $p$**, or just **an excluded point topology**.

The topological space $T = \left({S, \tau_{\bar p}}\right)$ is called the **excluded point space on $S$ by $p$**, or just **an excluded point space**.

### Finite Excluded Point Topology

Let $S$ be finite.

Then $\tau_{\bar p}$ is a **finite excluded point topology**, and $\left({S, \tau_{\bar p}}\right)$ is a **finite excluded point space**.

### Infinite Excluded Point Topology

Let $S$ be infinite.

Then $\tau_{\bar p}$ is an **infinite excluded point topology**, and $\left({S, \tau_{\bar p}}\right)$ is a **infinite excluded point space**.

## Also see

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

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 13 - 15$