Preliminaries: A metric space is a set (in my case I have multiple of these sets $o_i$) with a given metric (in my case $m$) defined on its elements, so in my case I have the metric spaces $s_i=(o_i, m)$. I am also given a family of function $w_i: o_i \rightarrow \mathbb R$, each with the respective $o_i$ defined as its domain.
Question 1: Now I would like to define the family of tuples $(s_i, w_i)$ or $(o_i, m, w_i)$. Is there a mathematical name for $(s_i, w_i)$ or $(o_i, m, w_i)$?
Question 2: Since, for my context, the $m$ is not too important, I would also be happy for a term of $(o_i, w_i)$. I know that I could just say it is a "function and its domain", but in my case the sets / the domains are the most important objects of consideration. The functions are just weighting functions over the elements of each respective set. That is why I am looking for name that includes both and possibly puts emphasis on the set.
Afterthought: Since $m$ is the same metric for each set, the union of all o_i is a metric space as well.