Set is Open iff Neighborhood of all its Points

Theorem
Let $\left({T, \vartheta}\right)$ be a topological space.

Let $V \subseteq T$ be a subset of $T$.

Then:
 * $V$ is an open set of $T$

iff:
 * $V$ is a neighborhood of all the points in $V$.

Proof

 * Let $V$ be open in $T$.

Let $z \in V$.

By definition, a neighborhood of $z$, is any subset of $T$ containing an open set which itself contains $z$.

But $V$ is itself an open set which itself contains $z$.

Hence by Subset of Itself, $V$ is a subset of $T$ which contains an open set which itself contains $z$.

So for all points of $z \in V$, $V$ is a neighborhood of $z$.


 * Now suppose that for all points of $z \in V$, $V$ is a neighborhood of $z$.

It can be seen that $V$ can be expressed as the union of all open sets contained in $V$.

Hence $V$ is open in $T$, by the axioms of a topological space.