Let $[a,b]$ be an interval. A partition of $[a,b]$ is $N+1$ points such that: $$P:= \{x_1= a, x_2, \ldots,x_{N+1}= b \}.$$
Moreover, for an interval $[a,b]$ we can take two partitions $P^1, P^2$, and a common refinement is called when we take the union of points in $P^1, P^2$ and sort these points. So the end product is also a partition of $[a,b]$.
My question is the following. Let $P^1$ be a partition of $[a,b]$ and $P^2$ be a partition of $[c,d]$ such that $[a,b] \cap [c,d] \neq \emptyset$. I want to take something like a common refinement with the sorted union of $P^1$ and $P^2$, but I cannot call this a common refinement anymore, because the union becomes a partition of $[a,d]$ (in other words, $P^1$ and $P^2$ are partitions of different intervals). Is there any formal definition for what I am trying to do here?
Not sure if there's a name for that kind of partition. Assuming the partition is for Riemann integration, one way to formalize your idea is as follows. Let $I_1:=[a,b]\subset\mathbb{R}$ and $I_2:=[c,d]\subset \mathbb{R}$ be two disjoint intervals, i.e. $I_1\cap I_2=\varnothing$. We can then define a double partition $P$ as any collection of $(N+M)$ points of the form $$P=\{a=x_1<x_2<\cdots<x_N=b\}\cup\{c=y_1<y_2<\cdots <y_M=d\}.$$ Then we say that $f:[a,b]\cup[c,d]\rightarrow\mathbb{R}$ is Riemann-integrable on $[a,b]\cup [c,d]$ if for every $\epsilon>0$, there exists a double partition $P$, with $$U(f,P)-L(f,P)<\epsilon$$ where $U$ and $L$ respectively denote upper and lower sums defined the natural way.