Definition:Excluded Point Topology

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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