Discrete Topology is Finest Topology

Theorem
Let $S$ be a set.

Let $\tau$ be the discrete topology on $S$.
 * $\tau$ is the finest topology on $S$.

Proof
Let $\tau$ be the discrete topology on $S$.

Then by definition $\tau = \mathcal P \left({S}\right)$, that is, is the power set of $S$.

Let $\phi$ be any topology on $S$.

Let $U \in \phi$.

Then, by the definition of topology, $U \subseteq S$.

Then, by the definition of power set, $U \in \mathcal P \left({S}\right)$.

Hence by definition of subset, $\phi \subseteq \tau$.

Hence by definition of finer topology, $\tau$ is finer than $\phi$.

Also see

 * Indiscrete Topology is Coarsest Topology