Product of Topological Manifolds is a Manifold

Last updated Nov 1, 2022

Let $M_{1}, \dots, M_{k}$ be Topological Manifolds of Manifold Dimension $n_{1}, \dots, n_{k}$. Then the Product Topological Space $M_{1} \times \cdots \times M_{k}$ is a Topological Manifold of Manifold Dimension $n_{1} + \cdots + n_{k}$.

We need to show that $M := M_{1} \times \cdots \times M_{k}$ is

1. Hausdorff
2. Second Countable
3. Locally Euclidean of Manifold Dimension $n:= n_{1} + \cdots + n_{k}$

To that end

1. Every Finite Product of Hausdorff Spaces is Hausdorff
2. Every Finite Product of Second Countable Spaces is Second Countable
3. Let $p := (p_{1}, \dots, p_{k}) \in M$ and let $p_{1} \in (U_{1}, \varphi_{1}), \dots, p_{k} \in (U_{k}, \varphi_{k})$ be Coordinate Charts for $M_{1}, \dots, M_{k}$ respectively. We claim $(U_{1} \times \cdots \times U_{k}, \varphi_{1} \times \cdots \times \varphi_{k}) =: (U, \varphi)$ is a Coordinate Chart of $M$ that contains $p$. By construction, $p \in U$. Furthermore, the Product Topological Space is generated by finite Cartesian Products of Open sets of the constituent spaces, so $U$ is Open. Because A Product of Homeomorphisms is a Homeomorphism, $\varphi$ is a Homeomorphism to $\hat{U_{1}} \times \cdots \times \hat{U_{k}} \subset \mathbb{R}^{n_{1}} \times \cdots \times \mathbb{R}^{n_{k}} \cong \mathbb{R}^{n}$. Since $p \in M$ was arbitrary, $M$ is Locally Euclidean of Manifold Dimension $n$. $\blacksquare$