Definition:Particular 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 $\tau_p$ of the power set $\mathcal P \left({S}\right)$ as:
- $\tau_p = \left\{{A \subseteq S: p \in A}\right\} \cup \left\{{\varnothing}\right\}$
... that is, all the subsets of $S$ which include $p$, along with the empty set.
Then $\tau_p$ is a topology called the particular point topology on $S$ by $p$, or just a particular point topology.
The topological space $T = \left({S, \tau_p}\right)$ is called the particular point space on $S$ by $p$, or just a particular point space.
Finite Particular Point Topology
Let $S$ be finite.
Then $\tau_p$ is a finite particular point topology, and $\left({S, \tau_p}\right)$ is a finite particular point space.
Infinite Particular Point Topology
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.
Also see
- Results about particular point topologies can be found here.
Sources
- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 8 - 10$