Talk:Supremum of Set Equals Maximum of Supremums of Subsets

Does this result specifically only hold for finite set of subsets? If not, would it be worth expanding to a general Definition:Indexed Family of Subsets? Same for Maximum of Supremums of Subsets Equals Supremum of Set(to be removed). --prime mover (talk) 15:29, 3 December 2015 (UTC)

For an infinite set of subsets, the infinite set {$\sup S_i$} may not be bounded above so that $S$ may not have a supremum as a real number. Therefore, this result is not directly extendable to the infinite case.

For Maximum of Supremums of Subsets Equals Supremum of Set(to be removed) the situation is different because the infinite set {$\sup S_i$} is bounded, so the possibility of extending that result to the infinite case should be good in my opinion.

I wouldn't know if it would be worth expanding to a general Definition:Indexed Family of Subsets. --Ivar Sand (talk) 09:56, 4 December 2015 (UTC)

Not to worry about families, then. But since the applicability of the results differs in the infinite case, that is worth making a statement about on $\mathsf{Pr} \infty \mathsf{fWiki}$. --prime mover (talk) 22:03, 4 December 2015 (UTC)

There is a theorem here that corresponds to the combination of infinite versions of the two theorems in question. It is called Supremum of Suprema and talks about ordered sets. I'll include a reference in both of the two theorems to Supremum of Suprema for the benefit of those readers that are interested in more general results. --Ivar Sand (talk) 08:52, 7 December 2015 (UTC)