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