Characterization of Open Set by Open Cover

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

Let $\UU$ be an open cover of $T$.

For each $U \in \UU$, let $\tau_U$ denote the subspace topology on $U$.

Let $W \subseteq S$.

Then $W$ is open is $T$ :
 * $\forall U \in \UU: W \cap U$ is open in $\struct{U, \tau_U}$