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 Set in Discrete Topology is Clopen every subset of $T$ is open.

Then by definition of subset:
 * $\mathcal B \subseteq \tau$

By definition of basis it is sufficient to prove:
 * $\forall U \in \tau: \forall x \in U: \exists B \in \mathcal B: x \in B \subseteq U$

Let $U$ be an open set of $T$.

Let $x \in U$.

Let $B = \left\{{x}\right\}$.