Neighborhood iff Contains Neighborhood

Theorem
Let $X$ be a topological space.

Let $x\in X$.

Let $V\subset X$ be a subset.

Then the following are equivalent:
 * $V$ is a neighborhood of $x$ in $X$
 * $V$ contains a neighborhood of $x$ in $X$

Proof
Follows directly from the definition of neighborhood and Subset Relation is Transitive.

Also see

 * Set is Open iff Neighborhood of all its Points