Is finite complement topology on $\mathbb{R^2}$ the sameas the product topology on $\mathbb{R}$ with the finite complement topology.If $\mathbb{R_{fc}}$ denote the finite complement toplogy in $\mathbb{R}$ the question are $\mathbb{R_{fc}} \times \mathbb{R_{fc}}=\mathbb{R_{fc}^2}$
I´m try find a contra example, but I fail, to this moment I tried $U=\mathbb{R}$, $V=\mathbb{R}-\lbrace 0 \rbrace$ and $U=\mathbb{Z}=V$ but it fail. If someone can give my a hint or in their case give me an particular example.
HINT: Let $V=\Bbb R\setminus\{0\}$. Is $V\times V$ open in $\Bbb R_\text{fc}\times\Bbb R_\text{fc}$? Is it open in $\left(\Bbb R^2\right)_\text{fc}$?