Talk:Interval of Ordered Set is Convex

How do we extend the notion of interval to tosets without confusion? --Dfeuer (talk) 09:17, 13 February 2013 (UTC)

I don't know. Admittedly I never use that notion of interval. Also, at the end of the day it's just an order-convex set, no? --Lord_Farin (talk) 09:20, 13 February 2013 (UTC)

It depends on exact definitions. The analogy to the reals gives a bounded order-convex set. I don't know what's the most common definition. --Dfeuer (talk) 09:24, 13 February 2013 (UTC)

The reals surely also admit unbounded intervals... --Lord_Farin (talk) 09:25, 13 February 2013 (UTC)

These beasts be slippery. $\mathsf{Pr} \infty \mathsf{fWiki}$ currently defines Definition:Real Interval/Open to be bounded, defines Definition:Real Interval/Unbounded Open to be what I would call an open ray, and defines a Definition:Real Interval, before going into all these things, to be something that excludes the possibility that it's unbounded. --Dfeuer (talk) 09:31, 13 February 2013 (UTC)

That is indeed the case - because when "real interval" is usually encountered, it is supposed that it is bounded at either end. Thus the default position. The case when one of the endpoints is $\pm \infty$ is less commonly encountered, at least in basic analysis. Hence the more specific definition first, the more general one following - and the most general one of all, where $\R$ itself is considered as a degenerate case of a real interval, added at the end to ensure completeness. As it stands, it works. I would counsel against structural amendments. --prime mover (talk) 12:39, 13 February 2013 (UTC)

I have taken the libery of amending Definition:Real Interval slightly to emphasize that the informal definition has multiple interpretations. This takes weight off the reading that the informal definition imposes any interval to be bounded. --Lord_Farin (talk) 12:59, 13 February 2013 (UTC)

I don't particularly have an opinion, mind you, except I like using similar terminology for similar things. If interval = convex set, then I won't throw a fit..... --Dfeuer (talk) 09:33, 13 February 2013 (UTC)

Interval Defined by Betweenness says it all. --Lord_Farin (talk) 12:59, 13 February 2013 (UTC)