Point in Discrete Space is Adherent Point

Theorem
Let $S$ be a set.

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

Let $U \subseteq S$.

Then $x$ is an adherent point of $U$ iff $x \in U$.

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 $U \subseteq S$.

From All Sets in Discrete Topology are Clopen it follows that $U$ is open in $T$.

Now, let $x \in S$.

It follows that any subset of $S$ containing $x$ is an open neighborhood of $x$.

Let $x \in U$.

Then any open neighborhood of $x$ contains (trivially) an element of $U$, that is, $x$.

So, by definition, $x$ is an adherent point of $\left\{{x}\right\}$.

Now suppose $x \notin U$.

Then $\left\{{x}\right\}$ is an open neighborhood of $x$ such that $\left\{{x}\right\} \cap U = \varnothing$.

So $x$ can not be an adherent point of $U$.