Task:Characteristic property of the coproduct topology

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Statement

Suppose $({ (X_\alpha,\mathcal{J}_\alpha) })_{ \alpha = I }^{ \infty }$ is an arbitrary collection of non-empty topological spaces, and $(Y,\mathcal{ K })$ is another topological space. Suppose $f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y$ is a map, then:

• $f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y$ is continuous if and only if for all $\alpha\in I$, $f\vert_{X_\alpha}:X_\alpha\rightarrow Y$ is continuous.

Proof

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