Task:Characteristic property of the coproduct topology

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Statement

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

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

Proof

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