Topological Space is Discrete iff All Points are Isolated

Theorem
Let $S$ be a set.

Let $\vartheta$ be the discrete topology on $S$.
 * All points in $S$ are isolated.

Proof
Let $T = \left({S, \vartheta}\right)$ be the discrete space on $S$.

Then by definition $\vartheta = \mathcal P \left({S}\right)$, that is, is the power set of $S$.

Let $x \in S$.

Then from All Sets in Discrete Topology are Clopen it follows that $\left\{{x}\right\}$ is open in $T$.

Then by definition $\left\{{x}\right\}$ is a neighborhood of $x$.

Finally we note that $x \in T$ is an isolated point of $T$ iff there exists a neighborhood of $x$ in $T$ containing no points other than $x$.

Hence the result.