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.

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.

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.

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