Definition:Excluded 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 $T = \left({S, \tau_{\bar p}}\right)$ be the excluded point space on $S$ by $p$.
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.
Countable Excluded Point Topology
Let $S$ be countably infinite.
Then $\tau_{\bar p}$ is a countable excluded point topology, and $\struct {S, \tau_{\bar p} }$ is a countable excluded point space.
Uncountable Excluded Point Topology
Let $S$ be uncountable.
Then $\tau_{\bar p}$ is an uncountable excluded point topology, and $\struct {S, \tau_{\bar p} }$ is an uncountable 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: $14 \text { - } 15$. Infinite Excluded Point Topology