If a set is open or closed in $\mathbb{R}$ is it also open or closed $\mathbb{R}^{n > 1}$?

Question

I have a homework question that I have to prove that a set is open in $\mathbb{R}^2$. I was wondering if it was enough to show that the set was open in $\mathbb{R}^2$ then say something like "since X was open in $\mathbb{R}$ then X is open in $\mathbb{R}^2$

Answer 1

As lulu well-marked, your statement isn't always true. As well, open sets in $\mathbb{R}^2$ and $\mathbb{R}$ are defined in different ways. In the product topology, $U$ is open in $\mathbb{R}^2$ if, and only if, it is the cartesian product of two open sets in $\mathbb{R}$. Therefore, open sets in $\mathbb{R}$ which can't be defined as such can't be open in $\mathbb{R}^2$. Any open interval $(a,b)$ in $\mathbb{R}$, for example, can't be open in $\mathbb{R}^2$, because it isn't defined as a product of open subsets of $\mathbb{R}$.