Basis for Discrete Topology

Theorem
Let $S$ be a set.

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

Let $\mathcal B$ be the set:
 * $\mathcal B := \left\{{\left\{{x}\right\}: x \in S}\right\}$

... that is, the set of all singleton subsets of $S$.

Then $\mathcal B$ is a basis for $T$.

Proof
Let $T = \left({S, \vartheta}\right)$ be the discrete space on $S$.

Let $U \in \vartheta$.

Then:
 * $\displaystyle U = \bigcup_{x \in U} \left\{{x}\right\}$

Hence:


 * $\forall x \in U: \exists \left\{{x}\right\} \in \mathcal B: \left\{{x}\right\} \subseteq U$

Thus $U$ is the union of elements of $\mathcal B$.

Hence by definition $\mathcal B$ is a basis for $T$.