# Definition:Included Set Topology

(Redirected from Definition:Included Set Space)

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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 $1$