Neighborhood Basis of Open Subspace iff Neighborhood Basis

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

Let $U \subseteq S$ be an open subset

Let $\tau_U$ denote the subspace topology on $U$.

Let $s \in U$.

Let $\NN \subseteq \powerset U$.

Then:
 * $\NN$ is a neighborhood basis of $s$ in $\struct{U, \tau_U}$ $\NN$ is a neighborhood basis of $s$ in $T$