Generalizing open subsets to $\Bbb{R}$ Prove that if $X$ and $Y$ are open subsets of $\Bbb{R}$ then $X\times Y$ is an open subset of $\Bbb{R}^2$. State and prove a generalization to $\Bbb{R}^n$.
I figure the best approach is just to start with the generalization. 
So if $X_1,...,X_n$ are open subsets of $\Bbb{R}^n$ then $X_1\times\cdots\times X_n$ are open subsets of $\Bbb{R}^n$. Prove this statement. 
This is where I'm struggling to figure out what to prove exactly. Do I use open balls? Or perhaps the complement? I solved a similar problem involving proving all subsets were closed.

Can you prove that if $(x_1, \ldots, x_n) \in O = (a_1,b_1) \times (a_2, b_2) \times \ldots \times (a_n, b_n)$ there exists $r > 0$ such that $B(x,r) \subset O$?
