Definition:Included Set Topology

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 $\mathcal P \left({S}\right)$ as:
 * $\tau_H = \left\{{A \subseteq S: H \subseteq A}\right\} \cup \left\{{\varnothing}\right\}$

... that is, all the subsets of $S$ which are supersets of $H$, along with the empty set $\varnothing$.

Then $\tau_H$ is a topology called the included set topology on $S$ by $H$, or just an included set topology.

The topological space $T = \left({S, \tau_H}\right)$ is called the included set space on $S$ by $H$, or just an included set space.

Also see

 * Particular Point Topology, an instance of an included set topology where $H = \left\{{p}\right\}$, a singleton.