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Let \(X_1\times \cdots \times X_n\) be a product space . For any topological space \(Y\) a function \(f:Y\to X_1\times \cdots X_n\) is continuous if and only if component function \(f_i=\pi_i\circ f\) is continuous, where \(\pi_i\colon X_1\times \cdots \times X_n\to X_i\) denotes the canonical projection . The product space topology is the unique topology on \(X_1\times \cdots \times X_n\) with this property.