In the definition of pseudo-Euclidean space, we have $\mathbb{R}^{p+q}$ with a non-degenerate quadratic form $Q$ where $p,q\geq 0$, $n\geq 1$ and $p+q=n$. For $x \in \mathbb{R}^{p+q}$ we have $Q(x_1,\dots,x_{p+q}) = (x_1^2+\dots+x_{p}^2) - (x_{p+1}^2 + \dots + x_{p+q}^2)$.
As a special case, what exactly are $\mathbb{R}^{0+1}$ and $\mathbb{R}^{1+0}$? Are those the same set?
In the first case the quadratic form is $−|x|^2$ (not very interesting...).
In the second one, it is the classical squared norm $|x|^2$.
Are they considered the same: no, because the second one is associated with dot product $xx'$ whereas there is no dot product associated with the first one.
Nevertheless, they generate the same topology.