Let $I$ be a set, and let $\{(X_k,\tau_k) : k \in I\}$ be a family of topological spaces. Suppose that there exist distinct indices $i,j \in I$ such that $X_i \neq \varnothing$ and $X_j \neq \varnothing$. Then the topological disjoint union $\bigsqcup_{k \in I} X_k$, equipped with the coproduct topology, is disconnected.