Graph (topological manifold)

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Definition

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

• $\Gamma(f):\eq\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^k\ \big\vert\ x\in U\wedge f(x)\eq y\}$

We claim that $\Gamma(f)$ is a topological $n$-manifold (literally a topological manifold of dimension $n$)

Furthermore, it has a global chart (a chart whose domain is the entire of $\Gamma(f)$):

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

Proof of claims

• Hausdorff property of a topological manifold - a subspace of a Hausdorff space is a Hausdorff space so "inherited" from $\mathbb{R}^{n+k}$
• Second countable topological space - a subspace of a second countable space is a second countable space so also inherited from $\mathbb{R}^{n+k}$
• Locally Euclidean of dimension $n$ - which we will now show
  • If we show that $(\Gamma(f),\varphi)$ is a homemorphism we're done.
    • Lemma 1: $\varphi:\Gamma(f)\rightarrow U$ is continuous
      • Consider $\pi:\mathbb{R}^n\times\mathbb{R}^k\rightarrow\mathbb{R}^n$ - the fist canonical projection we get from the product topology
        • By the characteristic property of the subspace topology (on $U$) we see that $\varphi$, which is the restriction of $\pi$ to $U$, is continuous.
        • This completes the proof

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