Product of Topological Manifolds

Question

If M and N are topological manifolds, then their product is a topological manifold.

I can show that the product manifold is locally Euclidean, but I'm not sure how to show it is Hausdorff and second countable.

Any help would be much appreciated.

Answer 1

The product of two second-countable topological spaces $X$ and $Y$ is always second countable, because if $U_i, V_j$ are bases for $X,Y$ then $U_i \times V_j$ is a basis for $X \times Y$.

Indeed, by definition of basis we want to prove that given a point $(x,y)\ in W$ where $W$ is an open set of $X \times Y$, we can find a basic open set $U_i \times V_j \subseteq W$ with $(x,y) \in U_i \times V_j$. By the definition of the product topology we can find open subsets $U,V$ of $X,Y$ such that $(x,y)\in U \times V \subseteq W$. Now use the second countability of $X$ and $Y$.

The product of two hausdorff spaces $X,Y$ is always hausdorff because if $(x_1,y_1),(x_2,y_2) \in X \times Y$ are distinct points then WLOG $x_1 \neq x_2$, so we can separate $x_1$ from $x_2$ using some disjoint open sets $U_1,U_2$ of $X$. Now $U_1 \times Y, U_2 \times Y$ separate the original points.