View source for Task:Characteristic property of the coproduct topology

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==Statement==
Suppose {{MSeq|(X_\alpha,\mathcal{J}_\alpha)|\alpha|I}} is an arbitrary collection of non-empty [[topological spaces]], and {{Top.|Y|K}} is another topological space. Suppose {{M|f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y}} is a [[map]], then{{rITTMJML}}:
* {{M|f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y}} is [[continuous]] {{iff}} for all {{M|\alpha\in I}}, {{M|f\vert_{X_\alpha}:X_\alpha\rightarrow Y }} is continuous.
==Proof==
{{Requires proof}}
==References==
<references/>
{{Theorem Of|Topology}}