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 $A \in \CC$.

$\blacksquare$

Sources

• 1989: Ryszard Engelking: General Topology (revised and completed ed.)

• Mizar article TOPGEN_2:13