Topology is Discrete iff All Singletons are Open

Theorem

Let $\struct {S, \tau}$ be a topological space.

Then:

$\tau$ is the discrete topology on $S$

if and only if:

$\forall x \in S: \set x \in \tau$

That is, if and only if every singleton of $S$ is $\tau$-open.

Proof

Sufficient Condition

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

Let $x \in S$ be arbitrary.

Then from Set in Discrete Topology is Clopen it follows directly that $\set x$ is open in $\struct {S, \tau}$.

$\Box$

Necessary Condition

Let $\struct {S, \tau}$ be such that:

$\forall x \in S: \set x \in \tau$

Let $T \subseteq S$ be arbitrary.

Then $T = \bigcup \set {\set t: t \in T}$.

We have by hypothesis that each $\set t$ is open in $\struct {S, \tau}$.

By definition of a topology, a union of open sets is open.

Hence $T$ is open in $\struct {S, \tau}$.

As $T$ is arbitrary, it follows by definition that $\tau$ is the discrete topology.

$\blacksquare$