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 $\powerset S$ 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.