Point in Discrete Space is Neighborhood

Theorem
Let $S$ be a set.

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

Let $x \in S$.

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

Proof
By definition, a neighborhood $N_x$ of $x$ is any subset of $S$ containing an open set which itself contains $x$.

That is:
 * $\exists U \in \vartheta: x \in U \subseteq N_x \subseteq S$

From All Sets in Discrete Topology are Clopen we have that $\left\{{x}\right\}$ is open set in $S$.

So from Subset of Itself, $\left\{{x}\right\}$ is a subset of $S$ containing an open set $\left\{{x}\right\}$ which itself contains $x$.

Hence the result.