Characteristic property of the disjoint union topology

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Finish proof at least, link to other pages

Statement

Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be a collection of topological spaces and let [ilmath](Y,\mathcal{ K })[/ilmath] be another topological space]]. We denote by [ilmath]\coprod_{\alpha\in I}X_\alpha[/ilmath] the disjoint unions of the underlying sets of the members of the family, and by [ilmath]\mathcal{J} [/ilmath] the disjoint union on it (so [ilmath](\coprod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] is the disjoint union topological construct of the [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] family) and lastly, let [ilmath]f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y[/ilmath] be a map (not necessarily continuous) then:
we state the characteristic property of disjoint union topology as follows:

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TODO: rewrite and rephrase this

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• [ilmath]f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y[/ilmath] is continuous if and only if [ilmath]\forall\alpha\in I\big[f\big\vert_{X_\alpha^*}:i_\alpha({X_\alpha})\rightarrow Y\text{ is continuous}\big][/ilmath]

Where (for [ilmath]\beta\in I[/ilmath]) we have [ilmath]i_\beta:X_\beta\rightarrow\coprod_{\alpha\in I}X_\alpha[/ilmath] given by [ilmath]i_\beta:x\mapsto(\beta,x)[/ilmath] are the canonical injections

Proof

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It's actually pretty easy, just got to be careful with the notation