# Discrete Topology is Finest Topology

## Theorem

Let $T = \left({S, \tau}\right)$ 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 $\phi$.

$\blacksquare$