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

Necessary Condition
This follows immediately from the definition of subspace topology.

Sufficient Condition
Let:
 * $W \cap U$ be open in $\struct{U, \tau_U}$ for each $U \in \UU$

From Open Set in Open Subspace:
 * $\forall U \in \UU : W \cap U$ is open in $T$

We have:

By :
 * $W$ is open in $T$