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.
$\blacksquare$