Definition:Included Set Topology
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Definition
Let $S$ be a set which is non-null.
Let $H \subseteq S$ be some subset of $S$.
We define a subset $\tau_H$ of the power set $\powerset S$ as:
- $\tau_H = \set {A \subseteq S: H \subseteq A} \cup \set \O$
that is, all the subsets of $S$ which are supersets of $H$, along with the empty set $\O$.
Then $\tau_H$ is a topology called the included set topology on $S$ by $H$, or just an included set topology.
The topological space $T = \struct {S, \tau_H}$ is called the included set space on $S$ by $H$, or just an included set space.
Also see
- Definition:Particular Point Topology, an instance of an included set topology where $H = \set p$, a singleton.
- Results about included set topologies can be found here.
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets: Exercise $1$