Basis for Discrete Topology

Theorem
Let $S$ be a set.

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

Let $\mathcal B$ be the set of all singleton subsets of $S$:
 * $\mathcal B := \set {\set x: x \in S}$.

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

Proof
Let $T = \struct {S, \tau}$ be the discrete space on $S$.

Let $U \in \tau$.

Then:
 * $\displaystyle U = \bigcup_{x \mathop \in U} \set x$

Hence:


 * $\forall x \in U: \exists \set x \in \mathcal B: \set x \subseteq U$

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

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