Definition:Particular Point Topology/Infinite
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Definition
Let $S$ be a set which is non-empty.
Let $p \in S$ be some particular point of $S$.
Let $T = \left({S, \tau_p}\right)$ be the particular point space on $S$ by $p$.
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.
Countable Particular Point Topology
Let $S$ be countably infinite.
Then $\tau_p$ is a countable particular point topology, and $\struct {S, \tau_p}$ is a countable particular point space.
Uncountable Particular Point Topology
Let $S$ be uncountable.
Then $\tau_p$ is an uncountable particular point topology, and $\struct {S, \tau_p}$ is an uncountable particular point space.
Also see
- Results about particular 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: $9 \text { - } 10$. Infinite Particular Point Topology