Help with the notation $(x,t)\in \mathbb R^n \times (0,\infty)$

Question

What is the meaning of $$(x,t)\in \mathbb R^n \times (0,\infty)\quad ?\tag 1\label1$$

I guess $x$ is a $n$-vector and $t$ is just a scalar, i.e. \begin{align} x&=(x_1, x_2, \dots, x_n)\in \mathbb R^n \tag 2\\ t&\in (0,\infty) \tag 3 \end{align}

Attempt 1:

Does \eqref{1} mean I have , i.e. \begin{align} (x_1, t), (x_2,t), \dots, (x_n,t) \tag 4 \end{align} I.e. $n$ number of points in $\mathbb R^2$ (I guess?).

Attempt 2:

Or does \eqref{1} mean \begin{align} (x_1, x_2, \dots, x_n,t) \tag 5 \end{align} I.e. just one point. But how many dimensions?

Answer 1

It is shorthand for the pair $(x, t)$ with $x \in \mathbb{R}^n, t \in (0, \infty)$, thus $$ ((x_1, \dotsc, x_n), t) $$ This nested tuple can be mapped to the flat tuple $$ (x_1, \dotsc, x_n, t) $$ of course.