Let \(U\subseteq \mathbb{R}^k\) be open and \(f\colon U\to \mathbb{R}^n\) is a continuous function. Then the graph \(\Gamma(f)=\{(x,f(x))\mid x\in U\}\subseteq \mathbb{R}^{n+k}\) (equipped with the subspace topology ) is a \(k\)-manifold .
Remarks
- The global chart is given by \((\Gamma(f), \varphi)\), where \(\varphi\colon \Gamma(f)\to U\) with \[ \varphi(x,y)=y, \quad (x,y)\in \Gamma(f). \] It is called graph coordinates.
Proof
Consider the function \(\phi(x)=(x,f(x))\).
This function is a topological embedding
onto \(\Gamma(f)\) and therefore \(\Gamma(f)\) is homeomorphic to \(U\) or locally Euclidean
.
Note, that Hausdorff property and second countability are satisfied, since both properties are inherited by subspaces.