Connected Subset of Union of Disjoint Open Sets

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

Let $A$ be a connected set of $T$.

Let $U, V$ be disjoint open sets.

Let $A \subseteq U \cup V$.

Then
 * either $A \subseteq U$ or $A \subseteq V$.

Proof
Let $U’ = A \cap U$ and $V’ = A \cap V$.

By definition $U’$ and $V’$ are open sets in the subspace $\struct{A, \tau_A}$.

From Intersection is Empty Implies Intersection of Subsets is Empty $U’$ and $V’$ are disjoint.

Hence $U’$ and $V’$ are separated sets by definition.

Now

Since $A$ is connected then one of $U’$ or $V’$ is empty.

assume that $V’ = \empty$.

Then