View source for Graph (topological manifold)

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==Definition==
Let {{M|U\in\mathcal{P}(\mathbb{R}^n)}} be an [[open set]], with {{M|\mathbb{R}^n}} denoting [[Euclidean n-space|Euclidean {{m|n}}-space]]. Let {{M|f:U\rightarrow\mathbb{R}^k}} be a [[continuous map]] and recall the {{link|graph|function}} of {{M|f}}, {{M|\Gamma(f)}} is defined as follows{{rITSMJML}}:
* {{M|\Gamma(f):\eq\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^k\ \big\vert\ x\in U\wedge f(x)\eq y\} }}<ref group="Note">This could surely be written:
* {{M|\Gamma(f):\eq\{(x,y)\in U\times\mathbb{R}^k\ \big\vert y\eq f(x)\} }}
Instead. {{XXX|Check this}}</ref>

We claim that {{M|\Gamma(f)}} is a [[topological n-manifold|topological {{n|manifold}}]] (literally a [[topological manifold]] of dimension {{M|n}})

Furthermore, it has a [[global chart]] (a [[chart]] whose domain is the entire of {{M|\Gamma(f)}}):
* {{M|(\Gamma(f),\varphi)}} with {{M|\varphi:\Gamma(f)\rightarrow U\subseteq\mathbb{R}^n}} by {{M|\varphi:(x,f(x))\mapsto x}}
==Proof of claims==
* [[Hausdorff]] property of a [[topological manifold]] - ''[[a subspace of a Hausdorff space is a Hausdorff space]]'' so "inherited" from {{M|\mathbb{R}^{n+k} }}
* [[Second countable topological space]] - ''[[a subspace of a second countable space is a second countable space]]'' so also inherited from {{M|\mathbb{R}^{n+k} }}
* [[Locally Euclidean of dimension n|Locally Euclidean of dimension {{n}}]] - which we will now show
** If we show that {{M|(\Gamma(f),\varphi)}} is a [[homemorphism]] we're done. 
*** '''''Lemma 1:''''' {{M|\varphi:\Gamma(f)\rightarrow U}} is [[continuous]]
**** Consider {{M|\pi:\mathbb{R}^n\times\mathbb{R}^k\rightarrow\mathbb{R}^n}} - the fist [[canonical projection]] we get from the [[product topology]]
***** By the ''[[characteristic property of the subspace topology]]'' (on {{M|U}}) we see that {{M|\varphi}}, which is the [[restriction]] of {{M|\pi}} to {{M|U}}, is continuous. 
***** This completes the proof
{{Requires proof|grade=A*|msg=Saving work, so check this. I found this tab at least 4 days after writing it}}
==Notes==
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==References==
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{{Definition|Smooth Manifolds|Manifolds|Topology|Topological Manifolds}}