What is an example of an open set A which we can write it in the form of a Cartesian product of two sets $B\times C$? (B and C not necessarily open)

Question

I know each open set not necessarily be the cartesian product of two open sets. but I want an example for this equation. Please help me. Thank you in advance!

Answer 1

$$B = (0,1)$$ $$C = (0,1)$$ $$A = B \times C = (0,1) \times (0,1)$$

It is (hopefully) clear that $B$ and $C$ are both open sets, since they are open intervals. To see why $A$ is open, show that every point in $A$ is an interior point.

Hint: Assume that arbitrary point in $A$ is $p = (x,y)$ where both $x$ and $y$ are in $(0,1)$. Now, try to find a ball around $p$ such that the whole ball is contained in $A$.

To answer your question in comments (whether every open set is cartesian product of two open sets):

No. For a simple counterexample, consider a open disk in $\mathbb{R}^2$. Do you see why this set cannot be Cartesian product of two open intervals in $\mathbb{R}$?