Components of an open set in product topology Suppose $U = U_1 \times U_2 \times \cdots \times U_n $ is an open set in the topological space $X = X_1 \times X_2 \times \cdots \times X_n $, equipped with the box topology. (Note that $n$ is finite here). Then are $U_1, U_2, \cdots U_n$ open?

Your statement is correct, and your proof is nearly correct.
Union of product is not product of union, as @positrón0802 has pointed out. Examples include putting two squares on plane, and observe that the union of two square can possible be not a square even if squares have intersection.
For your statement, since $U = \cup_\alpha B_1^\alpha \times \dots B_n^\alpha$, define $V=\cup_\alpha B_1^\alpha$. $V$ is an open set. Now observe that $V=U_1$.
