Definition:Excluded Set Topology
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Definition
Let $S$ be a set which is non-empty.
Let $H \subseteq S$ be some subset of $S$.
We define a subset $\tau_{\bar H}$ of the power set $\powerset S$ as:
- $\tau_{\bar H} = \set {A \subseteq S: A \cap H = \O} \cup \set S$
that is, all the subsets of $S$ which are disjoint from $H$, along with the set $S$.
Then $\tau_{\bar H}$ is a topology called the excluded set topology on $S$ by $H$, or just an excluded set topology.
The topological space $T = \struct {S, \tau_{\bar H} }$ is called the excluded set space on $S$ by $H$, or just an excluded set space.
Also see
- Results about excluded set 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: $13 \text { - } 15$. Excluded Point Topology: $7$