A question about open set.

Question

Suppose we have for $j=1,\dots, n+1$ the sets $$U_j=\{x\in\mathbb{R}^{n+1}\;|\;x_j>0\}.$$

I have to prove that they are open subsets of $\mathbb{R}^{n+1}$.

If I write $$U_j=\mathbb{R}^{n+1}\cap(\mathbb{R}\times\cdot\cdot\cdot\times[\mathbb{R}\cap(0,+\infty)]\times\cdot\cdot\cdot\times\mathbb{R}),$$ where $(\mathbb{R}\times\cdot\cdot\cdot\times[\mathbb{R}\cap(0,+\infty)]\times\cdot\cdot\cdot\times\mathbb{R})$ is a vector of $n+1$ components and $\mathbb{R}\cap (0,+\infty)$ is the jth component

It's correct? If yes, Is there an alternative proof?

Thanks!

Answer 1

The idea is correct, but your notation is not very appropriate. You can simply write that$$U_j=\mathbb R\times\mathbb R\times\cdots\times\mathbb R\times(0,\infty)\times\mathbb R\times\cdots\times\mathbb R.$$So, since it is the Cartesian product of open sets, it is an open set.

Answer 2

If you simply say $ x \in U_j $ then $ x_j > 0 $

Therefore for all points y inside sphere $ | y - x | < x_j / 2 $

there will be $ y_j > x_j / 2 > 0 $ (it's a good exercise to proof that), i.e.

$ \{ y: | y - x | < x_j / 2 \} \subset U_j $

Therefore $ U_j $ is open set by definition