# Proposition 5.18 - Tu's Introduction to Manifold

I was trying to prove the following proposition, which is left as exercise in Tu's Introduction To Manifolds.

Proposition 5.18 (An atlas for a product manifold). If $$\left\{ (U_{\alpha}, \phi_{\alpha}) \right\}$$ and $$\left\{ (V_i, \psi_i) \right\}$$ are $$C^{\infty}$$ atlases for the manifolds $$M$$ and $$N$$ of dimension $$m$$ and $$n$$, respectively, then the collection $$\left\{(U_\alpha \times V_i, \phi_{\alpha} \times \psi_i : U_{\alpha} \times V_i \to \mathbb{R}^m \times \mathbb{R}^n \right\}$$ of charts is a $$C^{\infty}$$ atlas $$M \times N$$. Therefore, $$M\times N$$ is a $$C^{\infty}$$ manifold of dimension $$m + n$$.

I provide also the definitions I believe I should use.

Definition 5.1. : A topological space $$M$$ is locally Euclidean of dimension $$n$$ if every point $$p$$ in $$M$$ has a neighborhood $$U$$ such that there's is a homeomorphism $$\phi$$ from $$U$$ onto an open subset of $$\mathbb{R}^n$$. We call the pair $$(U,\phi : U \to \mathbb{R}^n)$$ a chart, $$U$$ a coordinate neighborhhod or a coordinate open set, and $$\phi$$ a coordinate map or coordinate system on $$U$$. We say that a chart $$(U,\phi)$$ is centered at $$p \in U$$ if $$\phi(p) = 0$$.

Definition 5.2 A topological manifold is Hausdorff, second countable, locally Euclidean space. It is said to be of dimension $$n$$ if it is locally Euclidean of dimension $$n$$.

I think the proposition is straightforward to prove, because the maps $$\phi_\alpha$$, and $$\psi_i$$ are homeomorphism then the product $$\phi_\alpha \times \psi_i$$ is also an homeomorphism on the product topology induced by the family $$U_\alpha \times V_i$$ (which is the generic open set). So the map $$\phi_\alpha \times \psi_i$$ maps the open $$U_\alpha \times V_i$$ into the product of two open sets $$\mathcal{O}_{\mathbb{R}^m} \times \mathcal{O}_{\mathbb{R}^n}$$ which is an open set by definition of product topology.

Am I missing anything, or is it simple as that?

It is that easy. The product $$M\times N$$ is Hausdorff and second countable for any two Hausdorff and second countable spaces $$M, N$$. And the product of two charts is a chart.