View source for Characteristic property of the disjoint union topology/Statement

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{{Requires references|grade=A|msg=Munkres maybe, Lee's manifolds, certainly}}
{{Stub page|msg=A rewrite, while not urgent, would be nice|grade=C}}
==Statement==
</noinclude>Let {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} be a collection of [[topological spaces]] and let {{Top.|Y|K}} be another topological space]]. We denote by {{M|\coprod_{\alpha\in I}X_\alpha}} the {{link|disjoint union|set|s}} of the underlying sets of the members of the family, and by {{M|\mathcal{J} }} the {{link|disjoint union|topology}} on it (so {{M|(\coprod_{\alpha\in I}X_\alpha,\mathcal{J})}} is the disjoint union topological construct of the {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} family) and lastly, let {{M|f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y}} be a [[map]] (not necessarily [[continuous]]) then:<br/>
we state the ''characteristic property'' of disjoint union topology as follows:
{{Todo|1=rewrite and rephrase this}}
* {{M|f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y}} is [[continuous]] {{iff}} {{M|1=\forall\alpha\in I\big[f\big\vert_{X_\alpha^*}:i_\alpha({X_\alpha})\rightarrow Y\text{ is continuous}\big]}}
Where (for {{M|\beta\in I}}) we have {{M|i_\beta:X_\beta\rightarrow\coprod_{\alpha\in I}X_\alpha}} given by {{M|i_\beta:x\mapsto(\beta,x)}} are the [[canonical injections]]<noinclude>
==Notes==
<references group="Note"/>
==References==
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{{Definition|Topology}}{{Theorem Of|Topology}}
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