Properties of Discrete Topology

Theorem
Let $S$ be a set.

Let $\vartheta$ be the discrete topology on $S$.

Then:
 * $\vartheta$ is indeed a topology on $S$;
 * $\vartheta$ is the finest topology on $S$.

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

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


 * $\vartheta$ is a topology on $S$:


 * $(1): \quad$ By definition of power set, $\varnothing \in \mathcal P \left({S}\right)$ and $S \in \mathcal P \left({S}\right)$.
 * $(2): \quad$ From Power Set with Union is a Monoid, $\mathcal P \left({S}\right)$ is closed under set union.
 * $(3): \quad$ From Power Set with Intersection is a Monoid, $\mathcal P \left({S}\right)$ is closed under set intersection.


 * $\vartheta$ is the finest topology on $S$:

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, $\phi \subseteq \vartheta$.

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