Superset of Neighborhood in Topological Space is Neighborhood

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

Let $x \in S$.

Let $N$ be a neighborhood of $x$ in $T$.

Let $N \subseteq N' \subseteq S$.

Then $N'$ is a neighborhood of $x$ in $T$.

That is:
 * $\forall x \in S: \forall N \in \NN_x: N' \supseteq N \implies N' \in \NN_x$

where $\NN_x$ is the neighborhood filter of $x$.

Proof
By definition of neighborhood:
 * $\exists U \in \tau: x \in U \subseteq N \subseteq S$

where $U$ is an open set of $T$.

By Subset Relation is Transitive:
 * $U \subseteq N'$

The result follows by definition of neighborhood of $x$.