# Set is Open iff Neighborhood of all its Points

## Theorem

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

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

Then:

$V$ is an open set of $T$
$V$ is a neighborhood of all the points in $V$.

## Proof

### Necessary Condition

Let $V$ be open in $T$.

Let $z \in V$.

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

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

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

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

$\Box$

### Sufficient Condition

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

That is, for all $z \in V$ there exists an open set $T_z \subseteq V$ of $T$ such that $z \in T_z$.

$\ds \bigcup_{z \mathop \in V} T_z \subseteq V$

as $\forall z \in V: T_z \subseteq V$.

If $z \in V$, then $z \in T_z$ by definition of $T_z$.

So:

$\ds z \in \bigcup_{z \mathop \in V} T_z$

Thus we also have:

$\ds V \subseteq \bigcup_{z \mathop \in V} T_z$

Hence by definition of set equality:

$\ds V = \bigcup_{z \mathop \in V} T_z$

Thus $V$ can be expressed as a union of open sets.

Hence $V$ is open in $T$, by Open Set Axiom $\paren {\text O 1 }$: Union of Open Sets of a topological space.

$\blacksquare$