Discrete Topology is Finest Topology

Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.
 * $\tau$ is the finest topology on $S$.

Proof
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 discrete topological space, $U \in \tau$.

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