Point in Discrete Space is Neighborhood

Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.

Let $x \in S$.

Then $\set x$ is a neighborhood of $x$ in $T$.

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

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

From Set in Discrete Topology is Clopen we have that $\set x$ is open set in $S$.

So by Set is Subset of Itself, $\set x$ is a subset of $S$ containing an open set $\set x$ which itself contains $x$.

Hence the result.