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. --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 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 . --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)