Basis for Discrete Topology

Theorem

Let $S$ be a set.

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

Let $\BB$ be the set of all singleton subsets of $S$:

$\BB := \set {\set x: x \in S}$.

Then $\BB$ is a basis for $T$.

Proof

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

Let $U \in \tau$.

Then:

$\ds U = \bigcup_{x \mathop \in U} \set x$

Hence:

$\forall x \in U: \exists \set x \in \BB: \set x \subseteq U$

Thus $U$ is the union of elements of $\BB$.

Hence by definition $\BB$ is a basis for $T$.

$\blacksquare$