I need help on the following problem, any responses would be greatly appreciated:
Let $M$ be a topological $m$-manifold and $N$ be a topological $n$-manifold. Prove that $M \times N$ is a topological $(m + n)$-manifold.
I know how to prove that $M \times N$ is Hausdorff and that it has a countable base for its topology (namely the product topology). However, I am unsure on how to prove that it is locally Euclidean I.e. It has an open cover by sets that are homeomorphic to open subsets of $\mathbb{R}^{m+n}$.
Best Answer
Every point $(p,q) \in M\times N$ has a product open set $U\times V$ where $U \subseteq M$ and $V \subseteq N$ are each open with $p \in U, q \in V$. Observe that we can pick $U$ and $V$ such that $U$ is homeomorphic to $\mathbb{R}^m$ and $V$ is homeomorphic to $\mathbb{R}^n$. If $f:U \to \mathbb{R}^m$ and $g:V \to \mathbb{R}^n$ are homeomorphisms, we can define $F: U \times V \to \mathbb{R}^{m+n}$ given by $F(x,y) = (f(x), g(y))$. We see immediately that $F$ is continuous (each component is continuous), and since $f^{-1}$ and $g^{-1}$ exist and are continuous, we find that $F^{-1}$ exists, is continuous, and is given by $F^{-1}(x,y) = (f^{-1}(x), g^{-1}(y))$.