View source for Disjoint union topology

{{Stub page|grade=A|msg=Grade A until it is more presentable.}}
==Definition==
Suppose {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} be an indexed family of [[topological spaces]] ''that are non-empty''{{rITTMJML}}, the ''disjoint union topology'' is a [[topological space]]:
* with underlying set {{M|\coprod_{\alpha\in I}X_\alpha}}, this is the [[disjoint union (set)|disjoint union of sets]], recall {{M|(x,\beta)\in\coprod_{\alpha\in I}X_\alpha\iff \beta\in I\wedge x\in X_\beta}} and
* The [[topology]] where {{M|U\in\mathcal{P}(\coprod_{\alpha\in I}X_\alpha)}} is considered [[open set|open]] ''if and only if'' {{M|1=\forall \alpha\in I[X_\alpha\cap U\in\mathcal{J}_\alpha]}}<ref group="Note">There's a very nasty abuse of notation going on here. First, note a set {{M|U}} is going to be a bunch of points of the form {{M|(x,\gamma)}} for various {{M|x}}s and {{M|\gamma}}s ({{M|\in I}}). There is no "canonical projection" FROM the product to the spaces, as this would not be a [[function]]!</ref> - be sure to notice the abuse of notation going on here.
{{Todo|Flesh out notes, mention subspace {{M|X_\alpha\times\{\alpha\} }} and such}}
'''Claim 1: ''' this is indeed a topology
{{Todo|Define the [[canonical injections of the disjoint union topology]] here}}
Note that [[the canonical injections of the disjoint union topology are topological embeddings]]

==[[Characteristic property of the disjoint union topology|Characteristic property]]==
{{:Characteristic property of the disjoint union topology/Statement}}
==Proof of claims==
{{Requires proof|msg=Actually surprisingly easy to prove, done on paper. page 1, 7/8/2016, Intro to top manifolds notes. Filed}}
==Notes==
<references group="Note"/>
==References==
<references/>
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{{Todo|Investigate the need to be non-empty, I suspect it's because the union "collapses" in this case, and the space wouldn't be a part of union}}