# Definition:Excluded 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 $\tau_{\bar p}$ of the power set $\powerset S$ as:

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

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 $T = \struct {S, \tau_{\bar p} }$ 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 $\struct {S, \tau_{\bar p} }$ 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

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $13 \text { - } 15$. Excluded Point Topology