Topology Discrete iff All Singletons Open

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Theorem

Let $(X, \tau)$ be a topological space.


Then $\tau$ is the discrete topology on $X$ iff:

For all $x \in X$: $\{ x \} \in \tau$

That is, iff every singleton of $X$ is $\tau$-open.


Proof

Forward Implication

Follows directly from Set in Discrete Topology is Clopen.

$\Box$


Reverse Implication

Suppose that:

For all $x \in X$: $\{ x \} \in \tau$

Let $S \subseteq X$.

Then $S = \bigcup \left\{{ \{ s \}: s \in S }\right\}$.

Then since each $\{ s \}$ is open, and a union of open sets is open, $S$ is open.

Since this holds for all $S \subseteq X$, $\tau$ is the discrete topology.

$\blacksquare$