Total variation for functions, Meaning of supremum as used here? On Wikipedia article, here: < on definition 1.1 there says, "where the supremum runs over the set of all partitions ..." AFAIK supremum is defined for a set not a collection of sets. What does author imply here by saying "supremum runs over the set of all partitions" ? Does he mean the supremum is applied to a set which is simply a union of all partitions ?

When writing $$\sup_P f(P)$$ you have to specify which set $P$ belongs to. Here, $P$ belongs to the set of all partitions of $[a,b]$.

To relate to what you said, taking $E$ to be the set $$E=\\{\sum_{i=0}^{n_P-1}\lvert f(x_{i+1}) - f(x_i)\rvert : P\mbox{ is a partition of } [a,b]\\}$$ then the supremum written on wikipedia equals $\sup E$.