Working in the reals, not the extended reals, I recently encountered several mathematicians, including my lecturer whose area of interest is topology, independently refer to $(-\infty,\infty),$ presumably since it is a closed set, as a "closed interval". I was surprised, since for example a tall preadolescent isn't necessarily a tall person.
Perhaps I could get some consensus from this community, if possible, regarding these opposing claims about the closed set $(-\infty,\infty):$
- $(-\infty,\infty)$is a closed interval, because a closed interval is an interval that is a closed set (i.e., that is topologically closed);
- $(-\infty,\infty)$is an open interval, because a closed interval is interval that includes both its endpoints (i.e., that has a minimum and a maximum).
Update
For what it's worth, the ISO 80000-2:2009 document explicitly calls $(a,b)$ an open interval and $[a,b]$ a closed interval.