Talk:Q78240777

All closed intervals are bounded

A closed interval in a poset (or a preordered set) $(X,\leq )$ is defined as a subset of the form $[a,b]=\{x\in X\colon a\leq x\leq b\}$; it is necesarily bounded since it has a lower bound $a$ and an upper bound $b$. The current distinction of closed interval (Q78240777) and closed interval (Q78244956) possibly comes from a misunderstanding that $[-\infty ,\infty ]$ is a closed interval of the real line. This is not true, because $[-\infty ,\infty ]$ is not even a subset of $\mathbb {R}$. Instead, $[-\infty ,\infty ]$ is the extended real line having $\mathbb {R}$ as its proper subset and is a closed interval (and hence a bounded subset) of itself. The two items should be merged. 慈居 (talk) 09:42, 23 August 2023 (UTC)

This item used to be about the closed interval on the real line (even if the labels might not have said so, links to external sources and the only formula present till recently solidified this understanding). By adding a formula based on a preordered set you generalized/changed the items meaning.

In the spirit of the stability of identifiers, I suggest to create a new item for that more general definition. Toni 001 (talk) 06:41, 25 August 2023 (UTC)

I am glad to see your reply. In my understanding, the item was about the concept of closed intervals in general, not specifically closed intervals on the real line. And the linked external sources treat different aspects of the same concept depending on the authors' preferences, targeted readers, how much time and energy the authors have got to have detailed treatments, how general treatment is needed for what they want to say next, etc.

The EoM article on intervals first defines intervals as subsets of the real line, and then goes on for intervals on arbitrary posets in general. The MathWorld article completely devotes to real intervals. The definition of intervals as real intervals is sufficient for the following texts about equivalence of intervals, (star) convexity and (path) connectedness. The nLab article first defines intervals on posets and introduces real intervals as examples. This is not weird since the readers of nLab are mainly those who are interested in category theory and thus are expected to be pretty much skilled at doing math. I believe that the EoM and nLab articles are counterexamples to your claim that "links to external sources and the only formula present till recently solidified this understanding".

Also I have provided reference in the entity that intervals are also defined in preordered sets. See Independence, Additivity, Uncertainty (Q121775923). The definition $[a,b]=\{x\in \mathbb {R} \colon a\leq x\leq b\}$ is only valid when the parent preordered set is the real line, it will be more appropriate to mark it as deprecated. But of course, real intervals are itself a notable concept in mathematics and we can definitely create an item for it and move the formula to this new item. 慈居 (talk) 08:10, 25 August 2023 (UTC)

Before any of your edits on this item, in this state, there were two links, one to Mathworld and one to ISO (via "described by source"), both concerned with real intervals. This makes this item about the real interval.

If any further sources about more general concepts can be linked to a more general item. Toni 001 (talk) 06:44, 28 August 2023 (UTC)

The item was quite incomplete at the moment and the two sources alone is not sufficient to tell what the item is about. Mathworld is an online reference for mathematics that is far from complete (compared to Wikipedia, which is also far from complete). The ISO reference only treats very restricted small portion of mathematics, somewhat biased toward analysis (e.g. no treatments of topology at all). My edits are a completion, not a generalization of the item. 慈居 (talk) 09:03, 28 August 2023 (UTC)

Perhaps I have the duty to notify you that I have updated the English, Chinese and, before having this discussion, Korean articles for interval (Q185148) to cover some necessary details. Even before my edits, all the three articles covered the case of w:totally ordered sets, the English article also the integer intervals (en, ko, zh), which means that it is dangerous to say that the item was merely about real intervals. 慈居 (talk) 14:34, 28 August 2023 (UTC)