Products of Euclidean Open Sets are Open

Let $p$ and $q$ be natural numbers and let $\mathbb{R}$ be the real numbers. Identify the Cartesian product of Euclidean spaces $\mathbb{R}^p\times\mathbb{R}^q$ with $\mathbb{R}^{p+q}$ by writing a pair $(\xi,\eta)$, with $\xi=(\xi_1,\dots,\xi_p)$ and $\eta=(\eta_1,\dots,\eta_q)$, as the point $x\in\mathbb{R}^{p+q}$ whose coordinates are

$$x_i=\xi_i\ \ (1\le i\le p),\qquad x_{p+j}=\eta_j\ \ (1\le j\le q);$$

this is the concatenation map, a bijection with the coordinatewise inverse described in claim 1 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space. For $U\subseteq\mathbb{R}^p$ and $V\subseteq\mathbb{R}^q$, the product $U\times V$ is regarded as a subset of $\mathbb{R}^{p+q}$ under this identification.

Then the following hold.

1. (The whole space) $\mathbb{R}^p$ is an open subset of $\mathbb{R}^p$.

2. (Products) If $U$ is open in $\mathbb{R}^p$ and $V$ is open in $\mathbb{R}^q$, then $U\times V$ is open in $\mathbb{R}^{p+q}$.