Expansion of Included Set Topology

Theorem
Let $S$ be a set.

Let $A_1 \subseteq S$ and $A_2 \subseteq S$.

Let $T_1 = \left({S, \tau_{A_1}}\right)$ and $T_2 = \left({S, \tau_{A_2}}\right)$ be included set spaces on $S$.

Then: where:
 * $T_1 \ge T_2 \iff A_1 \subseteq A_2$
 * $T_1 > T_2 \iff A_1 \subsetneq A_2$
 * $T_1 \ge T_2$ denotes that $T_1$ is finer than $T_2$
 * $T_1 > T_2$ denotes that $T_1$ is strictly finer than $T_2$.