Set of Singletons is Smallest Basis of Discrete Space

Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.

Let $\BB = \set {\set x : x \in S}$.

Then $\BB$ is the smallest basis of $T$.

That is:
 * $\BB$ is a basis of $T$

and:
 * for every basis $\CC$ of $T$, $\BB \subseteq \CC$.

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

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

Let $\CC$ be a basis of $T$.

Let $A \in \BB$.

By definition of the set $\BB$:
 * $\exists x \in S: A = \set x$

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

Then by Singleton of Element is Subset:
 * $\set x \subseteq B$

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

Thus $A \in \CC$.