Graph (topological manifold)

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Let [ilmath]U\in\mathcal{P}(\mathbb{R}^n)[/ilmath] be an open set, with [ilmath]\mathbb{R}^n[/ilmath] denoting Euclidean [ilmath]n[/ilmath]-space. Let [ilmath]f:U\rightarrow\mathbb{R}^k[/ilmath] be a continuous map and recall the graph of [ilmath]f[/ilmath], [ilmath]\Gamma(f)[/ilmath] is defined as follows[1]:

  • [ilmath]\Gamma(f):\eq\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^k\ \big\vert\ x\in U\wedge f(x)\eq y\} [/ilmath][Note 1]

We claim that [ilmath]\Gamma(f)[/ilmath] is a topological [ilmath]n[/ilmath]-manifold (literally a topological manifold of dimension [ilmath]n[/ilmath])

Furthermore, it has a global chart (a chart whose domain is the entire of [ilmath]\Gamma(f)[/ilmath]):

  • [ilmath](\Gamma(f),\varphi)[/ilmath] with [ilmath]\varphi:\Gamma(f)\rightarrow U\subseteq\mathbb{R}^n[/ilmath] by [ilmath]\varphi:(x,f(x))\mapsto x[/ilmath]

Proof of claims

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  1. This could surely be written:
    • [ilmath]\Gamma(f):\eq\{(x,y)\in U\times\mathbb{R}^k\ \big\vert y\eq f(x)\} [/ilmath]
    TODO: Check this


  1. Introduction to Smooth Manifolds - John M. Lee