Set of Singletons is Smallest Basis of Discrete Space

Theorem
Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Let $\mathcal B = \left\{{\left\{{x}\right\} : x \in S}\right\}$.

Then $\mathcal B$ is the smallest basis of $T$.

That is:
 * $\mathcal B$ is a basis of $T$

and:
 * for every basis $\mathcal C$ of $T$, $\mathcal B \subseteq \mathcal C$.

Proof
By Basis for Discrete Topology $\mathcal B$ is a basis of $T$.

It remains to be shown that $\mathcal B$ is the smallest basis of $T$.

Let $\mathcal C$ be a basis of $T$.

Let $A \in \mathcal B$.

By definition of the set $\mathcal B$:
 * $\exists x \in S: A = \left\{{x}\right\}$

By definition of basis:
 * $\exists B \in \mathcal C: x \in B \subseteq A$

Then by Singleton of Element is Subset:
 * $\left\{{x}\right\} \subseteq B$

Hence $B = A$ by definition of set equality.

Thus $A \in \mathcal C$.