# Neighborhood iff Contains Neighborhood

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## 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.

$\blacksquare$