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Suppose \(\coprod_{\alpha\in A} X_\alpha\) is a disjoint union space . For any topological space \(Y\) a function \(f:Y\to \coprod_{\alpha\in A} X_\alpha} \) is continuous if and only if its restriction to each \(X_\alpha\) is continuous. The disjoint union space topology is the unique topology on \(\coprod_{\alpha\in A} X_\alpha\) with this property.